Pythagoreanism

(1) Pythagoreanism is the philosophy of the ancient Greek
philosopher
Pythagoras
(ca. 570 – ca. 490
BCE), which prescribed a highly structured way of life and espoused the
doctrine of metempsychosis (transmigration of the soul after death into
a new body, human or animal).

(2) Pythagoreanism is the philosophy of a group of philosophers
active in the fifth and the first half of the fourth century BCE, whom
Aristotle refers to as “the so-called Pythagoreans” and to
whom Plato also refers. Aristotle's expression, “so-called
Pythagoreans,” suggests both that at his time this group of
thinkers was commonly called Pythagoreans and, at the same time, calls
into question the actual connection between these thinkers and
Pythagoras himself. Aristotle ascribes no specific names to these
Pythagoreans, but the philosophy which he assigns to them is very
similar to what is found in the fragments of
Philolaus
of Croton (ca. 470-ca. 390 BCE).
Thus, Philolaus and his successor Eurytus are likely to have been the
most prominent of these Pythagoreans. Philolaus posits limiters and
unlimiteds as first principles and emphasizes the role of number in
understanding the cosmos. Aristotle also identifies a distinct group of
these so-called Pythagoreans who formulated a set of basic principles
known as the table of opposites. Plato's sole reference to Pythagoreans
cites their search for the numerical structure of contemporary music
and is probably an allusion to
Archytas
(ca.
420-ca. 350 BCE), who is the first great mathematician in the
Pythagorean tradition. Starting from the system of Philolaus he
developed his own sophisticated account of the world in terms of
mathematical proportion.

(3) Many other sixth-, fifth- and fourth-century thinkers are labeled
Pythagoreans in the Greek tradition after the fourth century BCE. By
the late fourth century CE many of the most prominent Greek
philosophers including Parmenides, Plato and Aristotle come to be
called Pythagoreans, with no historical justification. There are
nonetheless a number of thinkers of the fifth and fourth century BCE,
who can legitimately be called Pythagoreans, although often little is
known about them except their names. The most important of these
figures is Hippasus. What criterion should be used to identify an
early figure as a Pythagorean is controversial and there is debate
about individual cases. Fourth-century evidence shows that
Pythagoreanism gave an unusually large role to women for an ancient
philosophhical school. It is likely that the Pythagorean communities
that practiced a way of life that they traced back to Pythagoras died
out in the middle of the fourth century BCE.

(4) The last manifestation of Pythagoreanism, Neopythagoreanism, has
been the most influential. Neopythagoreanism is not a unified school of
thought but rather a tendency, stretching over many centuries, to view
Pythagoras, with no historical justification, as the central and
original figure in the whole Greek philosophical tradition. This
Pythagoras is often thought to have received his philosophy as a divine
revelation, which had been given even earlier to wise men of the
ancient Near East such as the Persian Magi, the Hebrews (Moses in
particular), and the Egyptian priests. All Greek philosophy after
Pythagoras, insofar as it may be true, is seen as derived from this
revelation. Thus, Plato's and Aristotle's ideas are viewed as derived
from Pythagoras (with the mediation of other early Pythagoreans). Many
pseudepigrapha are produced in later times in order to provide the
Pythagorean “originals” on which Plato and Aristotle drew.
Some strands of the Neopythagorean tradition emphasize Pythagoras as
master metaphysician, who supposedly originated what are, in fact, the
principles of Plato's later metaphysics, the one and the indefinite
dyad. Other Neopythagoreans celebrate Pythagoras as the founder of the
quadrivium of mathematical sciences (arithmetic, geometry,
astronomy and music), while still others portray him as a magician or
as a religious expert and sage, upon whom we should model our lives.
Neopythagoreanism probably began already in the second half of the fourth
century BCE among Plato's first successors in the Academy, but
particularly flourished from the first century BCE until the end of
antiquity. Neopythagoreanism has close connections to Middle and
Neoplatonism and from the time of Iamblichus (4th c. CE) is largely
absorbed into Neoplatonism. It was the Neopythagorean version of
Pythagoreanism that dominated in the Middle Ages and Renaissance.

In the ancient sources, Eurytus is most frequently mentioned in the
same breath as Philolaus, and he is probably the student of Philolaus
(Iamblichus, VP 148, 139). Aristoxenus (4th c. BCE) presents
Philolaus and Eurytus as the teachers of the last generation of
Pythagoreans (Diogenes Laertius VIII 46) and Diogenes Laertius reports
that Plato came to Italy to meet Philolaus and Eurytus after the death
of Socrates (III 46). In order to be the pupil of Philolaus, who was
born around 470, and teach the last generation of Pythagoreans around
400, Eurytus would need to be born between 450 and 440. The sources are
very confused as to which S. Italian city he was from, Croton
(Iamblichus, VP 148), Tarentum (Iamblichus, VP 267;
Diogenes Laertius VIII 46) or Metapontum (Iamblichus, VP 266
and 267). It may be that the Eurytus from Metapontum is a different
Eurytus. It is possible that Archytas studied with Eurytus, since
Theophrastus (Aristotle's successor in the Lyceum) cites Archytas as
the source for the one testimony we have about the philosophy of
Eurytus (Metaph. 6a 19–22). In the catalogue of Pythagoreans
at the end of Iamblichus' On the Pythagorean Life (267),
Eurytus appears between Philolaus and Archytas in the list of
Pythagoreans from Tarentum, which may thus suggest that he was regarded
as the pupil of Philolaus and a teacher of Archytas.

According to Theophrastus (Metaph. 6a 19–22), Eurytus
arranged pebbles in a certain way in order to show the number which
defined things in the world, such as a man or a horse. Aristotle refers
to the same practice (Metaph. 1092b8 ff.), and Alexander
provides commentary on the Aristotelian passage (CAG I.
827.9). Aristotle introduces Eurytus as someone who regarded numbers as
causes of substances by being the points that bound spatial magnitudes.
He says that Eurytus made likenesses of the shapes of things in the
natural world with pebbles and thus determined the number which belongs
to each thing by the number of pebbles required. Scholars often treat
Eurytus' procedure as puerile and have sometimes not taken him
seriously (Kahn 2001, 33), or suggested that Theophrastus is ironical
in his presentation (e.g., Zhmud 2012, 410-411). There is, however, no obvious irony in
Theophrastus' remarks. He, in fact, presents Eurytus very
positively as someone who showed in detail how specific parts of the
cosmos arose out of basic principles, in contrast to other thinkers,
who posit basic principles but do not go very far in explaining how the
world arises from those principles. This positive presentation may
reflect Theophrastus' source, Archytas, who perhaps saw Eurytus
as attempting to carry out Philolaus' project of determining the
numbers that give us knowledge of things in the world (Huffman 2005,
55; see also Netz 2014, 173-178).

How are we, then, to understand Eurytus' procedure? It does not seem
plausible to suppose that he simply drew a picture or an outline
drawing of a man or a horse and then counted the number of pebbles
required to make the outline (Riedweg 2005, 86) or fill in the
picture, since the number would vary with the size of the drawing and
the size of the pebbles. A large picture of a man would require many
more pebbles than a small one, so that it would seem arbitrary which
number to associate with man. This interpretation treats Eurytus as a
mosaicist and is largely derived from Alexander's
testimony. Aristotle's presentation supports another
interpretation. He draws a parallel with those who arrange numbers of
pebbles into shapes, such as a triangle or a square. This suggests
that Eurytus had observed that, e.g., any three points in a plane
determine a triangle and any four a quadrilateral. He may then have
drawn the general conclusion that any shape or structure was
determined by a unique number of points and tried to represent these
by setting out the necessary number of pebbles. Thus, the complex
structure of a three-dimensional object such as the human body would
require a large number of points, but the number of points required to
determine a human being could be expected to be unique and to differ
from the number that determined any other object in the natural world,
such as a horse (Kirk and Raven 1957, 313 ff.; Guthrie 1962, 273 ff.;
Barnes 1982, 390–391; Cambiano 1998). It is important to note
that nothing in these reports suggests that Eurytus thought that
things were composed of numbers or that he regarded the points that
defined a given thing as atoms of which things were made, as has
sometimes been supposed (Cornford 1922–1923,
10–11). Instead, he is best understood as making a bold attempt
to show that the structure of all things is determined by number and
thus to provide specifics for Philolaus' general thesis that all
things are known through number. Another approach is to argue that no
reference is being made to creating a picture out of pebbles. The
pebbles refer instead to counters on an abacus, which the Greeks used
for calculations. In this case Eurytus can be supposed to have started
by identifying certain basic numerical properties with features of the
world and then deriving the number of man or horse through
calculations using the abacus (Netz 2014, 173-178).

Aristotle refers to the Pythagoreans frequently in his extant works,
especially in the Metaphysics. There are several puzzles
about these references. First, his usual practice is to refer to the
Pythagoreans as a group rather than naming individuals. He mentions
Philolaus and Eurytus by name only once each and Archytas four
times. Yet, the basic Pythagorean system which he describes in most
detail in Metaphysics 1.5 shows such strong similarities to
the fragments of Philolaus that Philolaus must be the primary source
(Huffman 1993, 28-94, Schofield 2012, 147), although some scholars
emphasize that Aristotle clearly did use other sources (Primavesi
2012, 255) and even that Philolaus, while perhaps the acme of
Pythagorean philosophy, might not have represented mainstream
Pythagoreanism thus explaining why Aristotle refers to the
Pythagoreans as a group rather than singling out Philolaus (McKirahan
2013). Second, he frequently refers to the Pythagoreans that he
discusses as the “so-called” Pythagoreans. Why does he add
the qualifying phrase “so-called?” This phrase indicates
not that these are false Pythagoreans in contrast to some other true
Pythagoreans but rather that this is the standard way of referring to
these people, it is what people call them; but the phrase also
indicates that Aristotle has reservations about the name. Aristotle is
expressing his doubts about how or whether these figures are connected
to Pythagoras himself, whom Aristotle regards as a wonder-working
founder of a way of life rather than as participating in the tradition
of Presocratic cosmology (Huffman 1993, 31-34). It could also be that
it is the very variety of sources that Aristotle is using that leads
him to recognize that there are quite different stages in the
develpment of Pythagoreanism and hence to wonder in what sense a
figure like Philolaus who is at the end of that development should
still be called a Pythagorean (Primavesi 2014).

The biggest puzzle, however, concerns the philosophical system that
Aristotle assigns to the Pythagoreans. For the purposes of his
discussion in the Metaphysics, he treats most Pythagoreans as
adopting a mainstream system in contrast to another group of
Pythagoreans whose system is based on the table of opposites (see
section 2.4). The central thesis of the mainstream system is stated in
two basic ways: the Pythagoreans say that things are numbers or that
they are made out of numbers. In his most extended account of the
system in Metphysics 1.5, Aristotle says that the
Pythagoreans were led to this view by noticing more similarities
between things and numbers than between things and the elements, such
as fire and water, adopted by earlier thinkers. The Pythagoreans thus
concluded that things were or were made of numbers and that the
principles of numbers, the odd and the even, are principles of all
things. The odd is limited and the even unlimited. Aristotle
criticizes the Pythagoreans for being so enamored of numerical order
that they imposed it on the world even where it was not suggested by
the phenomena. Thus appearances suggested that there were nine
heavenly bodies orbiting in the heavens but, since they regarded ten
as the perfect number, they supposed that there must be a tenth
heavenly body, the counter-earth, which we cannot see. Later,
Aristotle is also critical of the Pythagoreans for employing
principles that do not derive from the sensible world, i.e.,
mathematical principles, even though all their efforts were directed
at explaining the physical world (Metaphysics 989b29). How
can they explain features of physical bodies such as weight or motion
using principles which have no weight and do not move (990a8-990a16)?
Indeed, it becomes clear that Aristotle interpreted the Pythagorean
cosmogony as starting out by constructing the number one. The one then
draws in the unlimited and produces the rest of the number series and
evidently the cosmos at the same time. The number one and the other
numbers from 1 to 10 are conceived of as physical entities
(Metaphysics 1091a13-18). The puzzle is that Aristotle's
description makes clear that he is basically describing Philolaus'
system (e.g., the counter-earth, limit and unlimited, the generation
of a one), yet a number of his central assertions are flatly
contradicted by the surviving fragments of Philolaus. Most
importantly, Philolaus never says that things are numbers or are made
out of numbers. For Philolaus things are composed of limiters and
unlimiteds held together by harmony (Frs. 1, 2 and 6) and unlimiteds
appear to include physical things like fire and breath (Fr. 7,
Aristotle Fr. 201). Numbers and the odd and the even do play a
prominent role in Philolaus (Frs. 4-5), but there is no hint that they
are understood as physical entites. Instead number has an
epistemological role: all things are known through number (Fr. 4). How
are we to explain this tension between what Aristotle reports and the
fragments of Philolaus? One approach is to recognize that Aristotle is
not giving a historical report of what the Pythagoreans said but an
interpretation of what he found in Philolaus and others. He does not
in fact know of any text in which the Pythagoreans said that things
were numbers or were made of numbers. Instead this is a conclusion
drawn by Aristotle; it is his summary statement of what the
Pythagorean system amounts to. That this is what Aristotle is doing is
suggested by another passage in the Metaphysics where he
starts out by flatly stating that the Pythagoreans say that all things
are numbers but then goes on to add “at least they apply
mathematical theories to bodies as if they (the bodies) consisted of
those numbers”(Metaphysics 1083b16). The “at
least” and “as if” show that Aristotle is drawing an
inference rather than referring to any explicit statement by the
Pythagoreans that things are numbers (Huffman 1993, 57-64). Thus for
Philolaus there are analogies between numbers and things and numbers
give us knowledge of things but Aristotle mistakenly takes this to be
equivalent to saying that things are numbers or are made of
numbers. Another approach is to argue that Aristotle was right that
Philolaus and other Pythagoreans thought of the number one and other
numbers as physical entities. The one constructed in Philolaus Fr. 7
is not just the primal physical unity but also the number one
(Schofield 2012). At the opposite extreme, Zhmud argues that Aristotle
has essentially invented this Pythagorean system with little regard
for what any actual Pythagoreans said in order to serve as background
for his account of Plato's theory of principles (2012a, 438,
394-414). Another approach tries to mitigate the differences between
Philolaus and Aristotle and suggests that Aristotle's emphasis on
number was derived from Pythagorean numerology that was independent of
Philolaus but that was combined with material from Philolaus as a
result of Aristotle's decision to present one mainstream Pythagorean
system (Primavesi 2012).

At Metaphysics 986a22, after presenting his account of the
philosophy of “the so-called” Pythagoreans (985b23), which
has strong connections to the philosophy of Philolaus, Aristotle turns
to “others of this same group” and assigns to them what is
commonly known as the table of opposites (the opposites arranged
according to column [kata sustoichian]). These Pythagoreans
presented the principles of reality as consisting of ten pairs of
opposites:

limit

unlimited

odd

even

unity

plurality

right

left

male

female

rest

motion

straight

crooked

light

darkness

good

bad

square

oblong

Aristotle then contrasts these Pythagoreans with Alcmaeon of Croton,
who said that the majority of human things come in pairs, and praises
the Pythagoreans for carefully defining the pairs of opposites both in
number and character, whereas Alcmaeon seemed to present a randomly
selected and ill-defined group of opposites. Aristotle suggests that
either Alcmaeon was influenced by these Pythagoreans or they by him.
Aristotle was thus not sure of the date of these Pythagoreans but
seems to entertain the idea that they either lived a little before
Alcmaeon or a little after, which would make them active anywhere from
the late 6th to the mid 5th century. Aristotle's manner of introducing
these Pythagoreans suggests that they are distinct from Philolaus and
his pupil Eurytus and perhaps earlier (Schofield 2012: 156), but it is
not possible to be more specific about their identity. It is possible
that Aristotle only knows of the table through oral transmission and
that there were no specific names attached to it.

The table shows a strong normative slant by including good in one
column and bad in the other. In contrast, while Philolaus posits the
first two opposites in the table, limit and unlimited, as first
principles, there is no suggestion in the extant fragments of Philolaus
that limit was good and unlimited bad. Opposites played a large role in
most Presocratic philosophical systems. The Pythagoreans who posited
the table of opposites differed from other early Greek philosophers not
only in the normative view of the opposites but also by including
strikingly abstract pairs such as straight and crooked and odd and
even, in contrast to the more concrete opposites such as hot and cold,
which are typical elsewhere in early Greek philosophy. Similar tables
of opposites appear in the Academy (Aristotle, Metaph.
1093b11; EN 1106b29 referring to Speusippus; Simplicius in
CAG IX. 247. 30ff.), and Aristotle himself seems at times to
adopt such a table (Metaph. 1004b27 ff.; Phys.
201b25). Later Platonists and Neopythagoreans will continue to develop
these tables (see Burkert 1972a, 52, n. 119 for a list). The table of
opposites thus provides one of the clearest cases of continuity
between early Pythagoreanism and Platonism. Zhmud argues that the
table has little to do with early Pythagoreanism and is largely a
product of the Academy (2012: 449-452), but Aristotle's discussion of
it in connection with Alcmaeon clearly shows that he regarded it as
belonging to the fifth-century and it is implausible to suppose that
he confused the work of his contemporaries in the Academy with
Pythagorean ideas that were developed over a century earlier. It may
well be that the similarity between this Pythagorean table of
opposites and later Academic versions led to the Neopythagorean habit,
starting already in the early Academy, of mistakenly assigning the
fundamental pair of opposites in Plato's late metaphysics, the one and
the indefinite dyad, back to Pythagoras (see on Neopythagoreanism
below).

Iamblichus' On the Pythagorean Life (4th c. CE) ends with a
catalogue of 218 Pythagorean men organized by city followed by a list
of 17 of the most famous Pythagorean women. Of these 235 Pythagoreans,
145 appear nowhere else in the ancient tradition. This impressive list
of names shows the wide impact of Pythagoreanism in the fifth and
fourth centuries BCE. To what extent is it reliable? A long line of
scholars has argued that the catalogue has close connections to and is
likely to be based on Aristoxenus in the fourth century BCE and is
thus a reasonably accurate reflection of early Pythagoreanism rather
than a creation of the later Neopythagorean tradition (Rohde
1871–1872, 171; Diels 1965, 23; Timpanaro-Cardini 1958-1964, III
38 ff.; Burkert 1972a, 105, n. 40; Zhmud 2012b, 235–244). This
is up to a point a reasonable conclusion, since it is hard to see who
would have been better placed than Aristoxenus to have such detailed
information.

The arguments connecting Aristoxenus to the catalogue are not
unassailable, however, and it is likely that the list has been altered
in transmission, so that it cannot simply be accepted as the testimony
of Aristoxenus (Huffman 2008a). No names on the list can be positively
assigned to a date later than Aristoxenus, but this would be likely to
be true, even if the list were compiled at a later date, since
Pythagoreanism appears to have largely died out for the two centuries
immediately following Aristoxenus' death. Thus, Iamblichus does not
mention any Pythagorean who can be positively dated after the time of
Aristoxenus anywhere else in On the Pythagorean Life
either. Scholars have also argued that Iamblichus cannot have composed
the catalogue, since he mentions some 18 names that do not appear in
the catalogue. This argument would only work, if Iamblichus were a
careful and systematic author, which the repetitions and
inconsistencies in On the Pythagorean Life show that he was
not. While it is unlikely that Iamblichus composed the catalogue from
scratch, it is perfectly possible that he edited it in a number of
ways, while not feeling compelled to make it consistent with
everything he says elsewhere in the text. There are some peculiarities
of the catalogue that suggest a connection to Aristoxenus. Philolaus
and Eurytus are listed not under Croton but under Tarentum, just as
they are in one of the Fragments of Aristoxenus (Fr. 19 Wehrli =
Diogenes Laertius VIII 46). On the other hand, some features of the
catalogue are inconsistent with what we know of
Aristoxenus. Aristoxenus' teacher, Xenophilus, who is identified as
from the Thracian Chalcidice in the Fragments of Aristoxenus (Frs. 18
and 19 Wehrli), is identified as from Cyzicus in the
catalogue. Moreover, the legendary figure, Abaris, is included in the
catalogue and even said to be from the mythical Hyperborea, whereas
Aristoxenus is usually seen as resolutely trying to rationalize the
Pythagorean tradition. Thus, while Aristoxenus is quite plausibly
taken to be the author of the core of the catalogue, it is likely that
additions, omissions, and various changes have been made to the
original document and hence it is impossible to be sure, in most
cases, whether a given name has the authority of Aristoxenus behind it
or not.

The catalogue includes several problematic names, such as Alcmaeon,
Empedocles, Parmenides and Melissus. Alcmaeon was active in Croton when
the Pythagoreans flourished there, but Aristotle explicitly
distinguishes Alcmaeon from the Pythagoreans and scholarly consensus is
that he is not a Pythagorean (see the entry
on Alcmaeon). Most
scholars would agree that Empedocles was heavily influenced by
Pythagoreanism; in the later tradition fragments of Empedocles are
routinely cited to support the Pythagorean doctrines of metempsychosis
and vegetarianism (e.g., Sextus Empiricus, Adversus
Mathematicos IX 126–30). On the other hand, both in the ancient
and in the modern world, Empedocles is not usually labeled a
Pythagorean, because, whatever the initial Pythagorean influences, he
developed a philosophical system that was his own original
contribution. Parmenides is again not usually identified as a
Pythagorean in either the ancient or modern tradition and, although
scholars have speculated on Pythagorean influences on Parmenides, there
is little that can be identified as overtly Pythagorean in his
philosophy. The reason for Parmenides' inclusion in the catalogue
is pretty clearly the tradition that his alleged teacher Ameinias was a
Pythagorean (Diogenes Laertius IX 21). There is no reason to doubt this
story, but it gives us no more reason to call Parmenides a Pythagorean
than to call Plato a Socratic or Aristotle a Platonist. It would appear
that Melissus was included on the list because he was regarded in turn
as the pupil of Parmenides. Inclusion in the catalogue thus need not
indicate that a figure lived a Pythagorean way of life or that he
adopted metaphysical principles that were distinctively Pythagorean; he
need only have had contact with a Pythagorean teacher. It is possible
that Aristoxenus included Parmenides and Melissus on the list for these
reasons or that he had better reasons for including them (e.g., evidence
that they lived a Pythagorean life), but it is precisely famous names
such as these that would be likely to have been added to the list in
later times, and they may well not have appeared in Aristoxenus'
catalogue at all.

Zhmud (2012a, 109–134) has argued that it begs the question to
use a doctrinal criterion to identify Pythagoreans. We need to first
identify Pythagoreans and then see what their doctrines
are. Aristoxenus' catalogue of Pythagoreans as preserved in Iamblichus
is the crucial source. Zhmud takes the Pythagoreans on this list whom
we can identify (the overwhelming majority are just names for us) and
studies their interests and activities in order to arrive at a picture
of early Pythagoreanism. Of the 235 names Zhmud finds only 15 about
whom we know anything significant. Some of these are non-controversial
(Hippasus, Philolaus, Eurytus and Archytas). However, Zhmud puts
particular emphasis on a series of figures not typically regarded as
Pythagoreans, e.g., Democedes, Alcmaeon, Iccus, Menestor,and
Hippon. The range of interests of these figures leads him to conclude
that there is no one characteristic that is shared by all Pythagoreans
and that Wittgestein's concept of a family resemblance should be
employed to describe Pythagoreanism. Moreover, his reliance on figures
like Alcmaeon and Menestor leads him to the surprising conclusion that
natural science and medicine were more important than mathematics for
the philosophical views of early Pythagoreans (2012a, 23). The
foundation for this view of early Pythagoreanism is problematic since
the scholarly consensus is that Alcmaeon was not a Pythagorean and it
is also far from certain that Menestor was a Pythagorean (see below).
As argued above, Iamblichus' catalogue cannot be used mechanically as
a guarantee that a given figure was a Pythagorean, because we cannot
be sure that it always reflects Aristoxenus. What criteria should then
be used (Huffman 2008a)?

First, anyone identified as a Pythagorean by an early source
uncontaminated by the Neopythagorean glorification of Pythagoras (see
below) can be regarded as a Pythagorean. This would include sources
dating before the early Academy (ca. 350 BCE), where Neopythagoreanism
has its origin, and Peripatetic sources contemporary with the early
Academy (ca. 350–300 BCE, e.g., Aristotle, Aristoxenus and Eudemus), who,
under the influence of Aristotle, defined themselves in opposition to
the Academic view of Pythagoras.

Second, a doctrinal criterion is applicable. Anyone who espouses the
philosophy assigned to the Pythagoreans by Aristotle can be regarded
as a Pythagorean, although Aristotle presents that philosophy under an
interpretation that must be taken into account. It is important that
the use of such a doctrinal criterion be limited to quite specific
doctrines such as limiters and unlimiteds as first principles and the
cosmology that includes the counter-earth and central
fire. Particularly to be avoided is the assumption that any early
mathematician or any early figure who assigns mathematical ideas a
role in the cosmos is a Pythagorean. Mathematicians such as Theodorus
of Cyrene (who is included in Iamblichus' catalogue) and Hippocrates
of Chios (who is not) are not treated as Pythagoreans in the early
sources such as Plato, Aristotle and Eudemus, and there is accordingly
no good reason to call them Pythagoreans. Similarly, the sculptor,
Polyclitus of Argos, stated that “the good comes to be …
through many numbers,” (Fr. 2 DK), but no ancient source calls
him a Pythagorean (Huffman 2002). As Burkert has emphasized,
mathematics is a Greek and not just a specifically Pythagorean passion
(1972a, 427).

Third, anyone universally (or almost universally) called a
Pythagorean by later sources, and whom early sources do not treat as
independent of Pythagoreanism, explicitly or implicitly, can be
regarded as a Pythagorean. This would include figures embedded in the
biographical tradition about Pythagoras and the early Pythagoreans,
such as the husband and wife, Myllias and Timycha.

This last criterion is more subjective than the first two and
difficult cases arise. The fifth-century botanist Menestor (DK I 375)
is discussed by Theophrastus and called one of “the old natural
philosophers” (CP VI 3.5) with no mention of any
Pythagoreanism. In this case, the inclusion of a Menestor in
Iamblichus' catalogue is not enough reason to regard Theophrastus'
Menestor as a Pythagorean. On the other hand, although Aristotle
treats Hippasus separately from the Pythagoreans, as he does Archytas,
the almost universal identification of Hippasus as a Pythagorean in
the later tradition and his deep involvement in the biography of early
Pythagoreanism, show that he should be regarded as a Pythagorean (on
Hippasus, see section 3.4 below). The fifth-century figure Hippo (DK I
385), who is derided by Aristotle and paired with Thales as positing
water as the first principle (Metaph. 984a3), is a
particularly difficult case. An Hippo is listed in Iamblichus'
catalogue under Samos and Censorinus tells us that Aristoxenus
assigned Hippo to Samos rather than Metapontum (DK I 385.4–5). This
makes it look as if Aristoxenus may be responsible for including Hippo
in the catalogue. Burkert has also tried to demonstrate connections
between Hippo's philosophy and that of the Pythagoreans (1972a, 290,
n. 62). On the other hand, neither Aristotle nor Theophrastus nor any
of the Aristotelian commentators call him a Pythagorean and the
doxographers describe this Hippo as from Rhegium (e.g., Hippolytus in
DK I 385.17). It is thus not clear whether we are dealing with one
person or two people named Hippo and it is doubtful that the Hippo
discussed by the Peripatetics was a Pythagorean (Zhmud regards Hippo
as well as Menestor and Theodorus as Pythagoreans — 2012a,
126–128). Those figures of the sixth, fifth and fourth century who have
the best claim to be considered Pythagoreans will be discussed in the
following sections.

In the standard collection of the fragments and testimonia of the
Presocratics, Cercops, Petron, Brontinus, Hippasus, Calliphon,
Democedes, and Parmeniscus are listed as older Pythagoreans (DK I
105–113). Hippasus, who is the most important of these figures, will be
discussed separately below (sect. 3.4). Of the rest only Brontinus,
Calliphon and Parmeniscus appear in Iamblichus' catalogue.

Brontinus is presented as either the husband or father of Theano (see
section 3.3 below). Brontinus (DK I 106–107) is elsewhere said
to have had a wife Deino and to be either from Metapontum or
Croton. The elusive connection between Orphism and Pythagoreanism
rears its head with Brontinus, since the fourth-century author,
Epigenes, reports that Brontinus is supposed to be the real author of
two works circulating in the name of Orpheus (West 1983, 9
ff.). Little else is known about him, but his existence appears to be
confirmed by Alcmaeon, writing in the late sixth or early fifth
century, who addresses his book to a Brontinus along with Leon and
Bathyllus (Fr. 1 DK). The latter two may also be Pythagoreans, since a
Leon is listed under Metapontum and a Bathylaus (sic) under
Posidonia, in Iamblichus' catalogue.

Cercops (DK I 105–106) is an even more obscure figure who is, again
according to Epigenes, the supposed Pythagorean author of Orphic texts
(West 1983, 9, 248), although Burkert doubts that he was a Pythagorean
(1972a, 130).

To Petron (DK I 106) is ascribed the startling doctrine that there
are 183 worlds arranged in a triangle, but he is only known from a
passage in Plutarch, is not called a Pythagorean there and is probably
a literary fiction (Guthrie 1962, 322–323; Burkert 1972a, 114; Zhmud
2012a, 117).

A Parmeniscus (DK I 112–113) is called a Pythagorean by Diogenes
Laertius (IX 20) and may be the same as the Parmiskos listed under
Metapontum in Iamblichus' catalogue. Athenaeus reports that a
Parmeniscus of Metapontum lost the ability to laugh after descending
into the cave of Trophonius, only to recover it in a temple on Delos,
where the surviving inventory of the temple of Artemis records a
dedication of a cup by a Parmiskos (Burkert 1972a, 154).

There no good reason to think that Democedes (DK I 110–112), the
physician from Croton, was himself a Pythagorean, although he had some
Pythagorean connections. He is famous from Herodotus' account
(III 125 ff.) of his service to the tyrant, Polycrates, and the Persian
king, Darius. One late source names him a Pythagorean (DK I 112.21).
Iamblichus mentions a Pythagorean named Democedes, who was involved in
the political turmoil surrounding the conspiracy of Cylon against the
Pythagoreans, but it is far from clear that this was the physician
(VP 257–261). Herodotus never calls Democedes a Pythagorean
nor do any other of the later sources (e.g., Aelian, Athenaeus, the
Suda), nor does he appear in Iamblichus' catalogue. A Calliphon,
who could be Democedes' father, is presented as an associate of
Pythagoras by Hermippus (DK I 111.36 ff.) and appears in
Iamblichus' catalogue, so it is reasonable to regard him as a
Pythagorean, although we know nothing more of him. It is reported
(Herodotus III 137) that Democedes married the daughter of the Olympic
victor, Milon, who was the Pythagorean, whose house was used as a
meeting place (Iamblichus, VP 249). It was undoubtedly because
Democedes came from Croton at roughly the time when Pythagoras was
prominent there and because of the Pythagorean connections of his
father and father-in-law that late sources came to label Democedes
himself a Pythagorean. For an argument that Democedes was a Pythagorean see Zhmud 2012a, 120.

Women were probably more active in Pythagoreanism than any other
ancient philosophical movement. The evidence is not extensive but is
sufficient to give us a glimpse of their role. At the end of the
catalogue of Pythagoreans in Iamblichus' On the Pythagorean
Life, after the list of 218 male Pythagoreans, the names of 17
Pythagorean women are given (VP 267). Since this list is
likely to be based on the work of Aristoxenus, it probably represents
what Aristoxenus learned from fourth-century Pythagoreans, although we
cannot, of course, be certain that some names were not inserted into
the list after the time of Aristoxenus (see section 3.1 above). Eleven
are identified as the wife, daughter or sister of a man but seven are
simply identified by their region or city-state of origin, although
the Echecrateia of Phlius listed seems likely to be connected to the
Echecrates of Phlius who appears in Plato's Phaedo. We know
nothing else about most of the names on the list and thus cannot be
sure in individual cases whether they belong to the sixth, fifth or
fourth century. For a speculative reconstruction of the role of women
in the Pythagorean society see Rowett (2014, 122–123), but this
reconstruction partly depends on the speech that Iamblichus reports
Pythagoras gave to the women of Croton upon his arrival (VP
54–57); however, while Pythagoras did give speeches to different
groups, including women, the text of the speech in Iamblichus is
probably a later fabrication (Burkert 1972a, 115; Zhmud 2012a,
70). The Pythagoreans put particular emphasis on marital fidelity on
the part of both men and women (Gemelli Marciano 2014, 145). There is
also no reliable evidence for any writings by these women, although in
the later tradition works were forged in the names of some of them and
of other Pythagorean women not on the list (see section 4.2
below).

The most famous name on the list is Theano who is here called the
wife of Brontinus but who is elsewhere treated as either the wife,
daughter or pupil of Pythagoras (Diogenes Laertius VIII 42; Burkert
1972a, 114). The role of women in early Pythagoreanism and the
centrality of Theano is further attested by Aristoxenus' contemporary,
Dicaearchus, who reports that Pythagoras had as followers not just men
but also women and that one of these, Theano, became famous (Fr. 40
Mirhday = Porphyry, VP 19). It is striking that Dicaearchus
does not identify her as the wife of either Brontius or Pythagoras but
simply as a follower of Pythagoras. In the later tradition a number of
works were forged in her name (see section 4.2 below), but we have
little reliable evidence about her (see Thesleff 1965, 193[special
character:ndash]201, for testimonia and texts; Delatte 1922,
246–249; and Montepaone 1993). The second most famous name on
the list is Timycha who, when ten months pregnant, reportedly bit off
her own tongue so that she could not, under torture, reveal
Pythagorean secrets to the tyrant Dionysius (Iamblichus, VP
189–194). This story goes back to Neanthes, writing in the late
fourth or early third century and may rely on local Pythagorean
tradition (Schorn 2014, 310).

Hippasus is a crucial figure in the history of Pythagoreanism,
because the tradition about him suggests that even in the fifth century
there was debate within the Pythagorean tradition itself as to whether
Pythagoras was largely important as the founder of a set of rules to
follow in living one's life or whether his teaching also had a
mathematical and scientific dimension. Hippasus was probably from
Metapontum (Aristotle, Metaph. 984a7; Diogenes Laertius VIII
84), although Iamblichus says there was controversy as to whether he
was from Metapontum or Croton (VP 81), and he is listed under
Sybaris in Iamblichus' catalogue (VP 267). He is
consistently portrayed as a rebel in the Pythagorean tradition, in one
case a democratic rebel who challenged the aristocratic Pythagorean
leadership in Croton (Iamb. VP 257), but more commonly as the
thinker who initiated Pythagorean study of mathematics and the natural
world.

It is in this latter role that he is connected with the split
between two groups in ancient Pythagoreanism, the acusmatici
(who emphasized rules for living one's life, including various taboos)
and the mathêmatici (who emphasized study of mathematics
and the natural world). Each group claimed to be the true Pythagoreans.
Our knowledge of this split comes from Iamblichus, who unfortunately
presents two contradictory versions, with the result that Hippasus is
sometimes said to be one of the mathêmatici and
sometimes one of the acusmatici. Burkert has convincingly
shown that the correct version is that reported by Iamblichus at De
Communi Mathematica Scientia 76.19 ff. (1972a, 192 ff.). According
to this account, the acusmatici denied that the
mathêmatici were Pythagoreans at all, saying that their
philosophy derived from Hippasus instead. The
mathêmatici for their part, while recognizing that the
acusmatici were Pythagoreans of a sort, argued that they
themselves were Pythagoreans in a truer sense. Hippasus' supposed
innovations, they said, were in fact plagiarisms from Pythagoras
himself. The mathêmatici explained that, upon
Pythagoras' arrival in Italy, the leading men in the cities did
not have time to learn the sciences and the proofs of what Pythagoras
said, so that Pythagoras just gave them instructions on how to act,
without explaining the reasons. The younger men, who did have the
leisure to devote to study, learned the mathematical sciences and the
proofs. The former group were the first acusmatici, who
learned the oral instructions of Pythagoras on how to live (the
acusmata = “things heard”), while the latter
group were the first mathêmatici. Hippasus was thus
closely connected to the mathêmatici in this split in
Pythagoreanism but ended up being disavowed by both sides. For an
attempt to further characterize the mathêmatici see
Horky 2013.

It is difficult to be sure of Hippasus' dates, but he is typically
regarded as active in the first half of the fifth century and perhaps
early in that period (Burkert 1972a, 206; Zhmud 2012a, 124-125). The
split in Pythagoreanism may have occurred after the main period of his
work and was perhaps connected to the attacks on the Pythagorean
societies by outsiders around 450 BCE (Burkert 1972a, 207), but
certainty is not possible. Zhmud (2012a, 169–192) has argued
that the split is an invention of the later tradition, appearing first
in Clement of Alexandria and disappearing after Iamblichus. He also
notes that the term acusmata appears first in Iamblichus
(On the Pythagorean Life 82–86) and suggests that it
also is a creation of the later tradition. He admits that the
Pythagorean maxims did exist earlier, as the testimony of Aristotle
shows, but they were known as symbola, were originally very
few in number and were mainly a literary phenomenon rather than being
tied to people who actually practiced them (Zhmud 2012a,
192–195). The consensus view, which accepts the split, is based
on Burkert's argument that Iamblichus'account of the split between
the acusmatici and mathêmatici can be shown to
be derived from Aristotle (1972a, 196). Burkert later reaffirmed this
position, although with a little less confidence, asserting that the
Aristotelian provenance of the text is “as obvious as it is
unprovable” (1998, 315). Indeed the description of the split in
what is likely to be the original version (Iamblichus, On General
Mathematical Science 76.16–77.18) uses language in
describing the Pythagoreans that is almost an Aristotelian signature,
“There are two forms of the Italian philosophy which is called
Pythagorean” (76.16). Aristotle famously describes the
Pythagoreans as “those called Pythagoreans” and also as
“the Italians” (e.g., Mete.
342b30, Cael. 293a20). Thus, Aristotle remains the most
likely source. Zhmud also argues against the split on the grounds that
there are no individuals in the historical record that can be
confidently identified as acusmatici. Since
the acusmatici were neither original nor full-time
philosophers, however, and simply preserved the oral taboos handed
down by Pythagoras, it is not surprising that they are not singled out
for attention in the sources. Only a relatively small number of the
names in Iamblichus' catalogue can certainly be identified
as mathêmatici and most of the others, particularly the
145 individuals whose names are only known from the catalogue, are
likely to be acusmatici, who to a greater or lesser degree
followed the Pythagorean
acusmata, but left no other trace of their activity. In
addition, a number of other Pythagoreans of the fifth and fourth
century, who figure in anecdotes about the Pythagorean life are likely
to be acusmatici (see below).

Hippasus is the first figure in the Pythagorean tradition who can
with some confidence be identified as a natural philosopher,
mathematician and music theorist. His connections are as much with
figures outside the Pythagorean tradition as those within it. This
independence may explain why neither Aristotle nor the doxographical
tradition label him a Pythagorean, but he is too deeply embedded in the
traditions about early Pythagoreanism for there to be any doubt that he
was in some sense a Pythagorean. Aristotle pairs Hippasus with
Heraclitus as positing fire as the primary element (Metaph.
984a7) and this pairing is repeated in the doxography that descends
from Theophrastus (DK I 109. 5–16), according to which Hippasus also
said that the soul was made of fire. Philolaus, who was probably two
generations later than Hippasus, might have been influenced by Hippasus
in starting his cosmology with the central fire (Fr. 7 Huffman). For
Philolaus, however, the central fire is a compound of limiter and
unlimited, whereas Hippasus is presented as a monist and does not start
from Philolaus' fundamental opposition between limiters and
unlimiteds.

There are only a few other assertions about the cosmology of
Hippasus and most of these seem to be the result of Peripatetic
attempts to classify him, such as the assertions that he makes all
things from fire by condensation and rarefaction and dissolves all
things into fire, which is the one underlying nature and that he and
Heraclitus regarded the universe as one, (always) moving and limited in
extent (DK I 109.8–10). More intriguing is the claim that he thought
there was “a fixed time for the change of the cosmos”
(Diogenes Laertius VIII 84), which might be a reference to a doctrine
of eternal recurrence, according to which events exactly repeat
themselves at fixed periods of time. This doctrine is attested
elsewhere for Pythagoras (Dicaearchus in Porphyry, VP 19). Our
information about Hippasus is sketchy, because he evidently did not
write a book. Demetrius of Magnesia (1st century BCE) reports that
Hippasus left nothing behind in writing (Diogenes Laertius VIII 84) and
this is in accord with the tradition that Philolaus was the first
Pythagorean to write a book (Huffman 1993, 15).

Hippasus originates the early Pythagorean tradition of scientific
and mathematical analysis of music, which reaches its culmination in
Archytas a century later. The correspondence between the central
musical concords of the octave, fifth, and fourth and the whole number
ratios 2 : 1, 3 : 2 and 4 : 3 is reflected in the acusmata
(Iamblichus, VP 82) and was thus probably already known by
Pythagoras. This correspondence was central to Philolaus'
conception of the cosmos (Fr. 6a Huffman). Although the later tradition
tried to assign the discovery to Pythagoras himself (Iamblichus,
VP 115), the method described in the story would not in fact
have worked (Burkert 1972a, 375–376). Hippasus is the first
person to whom is assigned an experiment demonstrating these
correspondences that is scientifically possible. Aristoxenus (Fr. 90
Wehrli = DK I 109. 31 ff.) reports that Hippasus prepared four bronze
disks of equal diameters, whose thicknesses were in the given ratios,
and it is true that, if free hanging disks of equal diameter are
struck, the sound produced by, e.g., a disk half as thick as another
will be an octave apart from the sound produced by the other disk
(Burkert 1972a, 377). Hippasus, thus, may be the first person to
devise an experiment to show that a physical law can be expressed
mathematically (Zhmud 2012a, 310).

Another text associates Hippasus with Lasus of Hermione in an
attempt to demonstrate the correspondence by filling vessels with
liquid in the appropriate ratios. It is less clear whether this
experiment would have worked as described (Barker 1989, 31–32). Lasus
was prominent in Athens in the second half of the sixth century at the
time of the Pisistratid tyranny and was thus probably a generation
older than Hippasus. There is no indication that Lasus was a
Pythagorean and this testimony suggests that the discovery of and
interest in the mathematical basis of the concordant musical intervals
was not limited to the Pythagorean tradition. Lasus and Hippasus are
sometimes said to have been the first to put forth the influential but
mistaken thesis that the pitch of a sound depended on the speed with
which it travels, but it is far more likely that Archytas originated
this view (Huffman 2005, 138–139). In the later tradition Hippasus is
reported to have ranked the musical intervals in terms of degrees of
concordance, making the octave the most concordant, followed by the
fifth, octave + fifth, fourth and double octave (Boethius,
Mus. II 19; see Huffman 2005, 433).

Finally, Iamblichus associates Hippasus with the history of the
development of the mathematics of means (DK I 110. 30–37), which are
important in music theory, but Iamblichus' reports are confused.
It is likely that Hippasus worked only with the three earliest means
(the arithmetic, geometric and subcontrary/harmonic) and that the
changing of the name of the subcontrary mean to the harmonic mean
should be ascribed to Archytas rather than Hippasus (Huffman 2005,
170–173).

The most romantic aspect of the tradition concerning Hippasus is the
report that he drowned at sea in punishment for the impiety of making
public and giving a diagram of the dodecahedron, a figure with twelve
surfaces each in the shape of a regular pentagon (Iamblichus,
VP 88). This is best understood as reflecting some sort of
mathematical analysis of the dodecahedron by Hippasus, but it is
implausible in terms of the history of Greek mathematics to suppose
that he carried out a strict construction of the dodecahedron, which
along with the other four regular solids is most likely to have first
received rigorous treatment by Theaetetus in the fourth century BCE
(Mueller 1997, 277; Waterhouse 1972; Sachs1917, 82). Nor is it clear
why public presentation of technical mathematical analysis should cause
a scandal, since few people would understand it. The most likely
explanation is that the dodecahedron was a cult object for the
Pythagoreans (dodecahedra in stone and bronze have been found dating
back to prehistoric times) and that it was because of these religious
connections that Hippasus' public work on the mathematical
aspects of the solid was seen as impious (Burkert 1972a, 460).

Another late story, which appears first in Plutarch, reports a scandal
which arose when knowledge of irrational magnitudes was revealed,
without specifying any punishment for the one who revealed it
(Numa 22). In Pappus' later version of the story, the person
who first spread knowledge of the existence of the irrational was
punished by drowning (Junge and Thomson 1930, 63–64). Iamblichus
knows two different versions of the story, one according to which the
malefactor was banished and a tomb was erected for him, signifying his
expulsion from the community (VP 246), but another according
to which he was punished by drowning as was the person (not
specifically said to be Hippasus here) who revealed the dodecahedron
(VP 247). Modern scholars have tried to combine the two
stories and suppose that Hippasus discovered the irrational through
his work on the dodecahedron (von Fritz 1945). This is pure
speculation, however, since neither does any ancient source connect
Hippasus to the discovery of the irrational nor does any source relate
the discovery of the irrational to the dodecahedron (Burkert 1972a,
459). Some scholars nonetheless credit Hippasus with the discovery of
irrationality (Zhmud 2012a, 274-278).

Some have argued that Hippasus was an important figure for the
early Academy to whom Academic doctrines were ascribed in order give
them his authority and even that he might be the Prometheus mentioned
by Plato as handing down the method from the gods in
the Philebus (Horky 2013). However, there is no explicit
mention of Hippasus by any member of the Academy and he is a minor
figure in fourth-century accounts of early Greek philosophy (e.g.,
Aristotle) so it is hard to see what authority he could give to
Academic views.

The other major Pythagoreans of the fifth century were Philolaus and
Eurytus, who are discussed above.

The name, but not too much more, is known of a number of other fifth
century figures, who with varying degrees of probability may be
considered Pythagoreans. To the beginning of the fifth century belongs
Ameinias the teacher of Parmenides (Diogenes Laertius VIII 21). The
athlete and trainer, Iccus of Tarentum, is listed in Iamblichus'
catalogue, but none of the other sources, including Plato, call him a
Pythagorean. In the later tradition, he was famous for the simplicity
of his life and “the dinner of Iccus” was proverbial for
plain fare. Plato praises his self control and reports that he touched
neither women nor boys while training. (Laws 839e; see
Protagoras 316d and DK I 216. 11 ff.).

Some scholars have treated the Sicilian comic poet Epicharmus as a
Pythagorean and argued that the growing argument which appears in a
fragment of controversial authenticity ascribed to him in Diogenes
Laertius (3.11) is thus Pythagorean in origin (Horky 2013,
131-140). However, no fifth- or fourth-century source identifies
Epicharmus as a Pythagorean and he does not appear in the catalogue of
Iamblichus. The earliest explicit mention of him as a Pythagorean is
in Plutarch (Numa 9) in the first century CE. There is no
compelling evidence that the reference to Epicharmus as a Pythagorean
in Iamblichus' On the Pythagorean Life 266 derives from the
fourth-century historian Timaeus as Horky proposes (2013,
116). Burkert suggests that the information on Didorus in 266 might
derive from Timaeus (1972, 203-204) but Iamblichus regularly combines
material from a number of sources so that neither Burkert nor most
scholars regard the passage as a whole as deriving from Timaeus
(Schorn 2014 only mentions VP 254-264 as having material from
Timaeus). Epicharmus has also been thought to be a Pythagorean because
the growing argument which he uses for comic effect uses pebbles to
represent numbers and refers to odd and even numbers. However, neither
of the features is peculiarly Pythagorean; the concept of odd and even
numbers belongs to Greek mathematics in general and not just to the
Pythagoreans and the use of counters (pebbles) on an abacus is the
standard way in which Greeks manipulated numbers (Netz 2014, 178;
cf. Burkert's doubts that there is anything Pythagorean in the
Epicharmus fragment 1972a, 438). Most scholars regard Epicharmus'
Pythagoreanism as a creation of the later tradition (Zhmud 2012a, 118;
Riedweg 2005, 115; Kahn 2001, 87).

There is no reason to regard the physician Acron of Acragas as a
Pythagorean, as Zhmud does (1997, 73; he appears to have changed his
mind in 2012a, 116). Acron is a contemporary of Empedocles and is
connected to him in the doxographical tradition (DK I 283. 1–9;
Diogenes Laertius VIII 65). No ancient source calls him a
Pythagorean. His name appears in a very lacunose papyrus along with
the name of Aristoxenus (Aristoxenus, Fr. 22 Wehrli), but it is pure
speculation that Aristoxenus labeled him a Pythagorean; Euryphon the
Cnidian doctor of the fifth century, who was not a Pythagorean, also
appears in the papyrus. Acron's father's name was Xenon, and a Xenon
appears in Iamblichus' catalogue, but he is listed as from Locri and
not Acragas, so again this is not good evidence that Acron was a
Pythagorean.

The Pythagorean Paron (DK I 217. 10–15) is probably a fiction
resulting from a misreading of Aristotle (Burkert 1972a, 170).
Aristotle reports the expression of a certain Xuthus, that “the
universe would swell like the ocean,” if there were not void into
which parts of the universe could withdraw, when compressed
(Physics 216b25). Simplicius says, on unknown grounds, that
this Xuthus was a Pythagorean, and scholars have speculated that he was
responding to Parmenides (DK I. 376. 20–26; Kirk and Raven 1957,
301–302; Barnes 1982, 616).

Aristoxenus reports that two Tarentines, Lysis and Archippus, were
the sole survivors when the house of Milo in Croton was burned, during
a meeting of the Pythagoreans, by their enemies (Iamblichus,
VP 250). A later romantic version in Plutarch (On the Sign
of Socrates 583a) has it that Lysis and Philolaus were the two
survivors, but it appears that the famous name of Philolaus has been
substituted for Archippus, about whom nothing else is known.
Aristoxenus goes on to say that Lysis left southern Italy and went
first to Achaea in the Peloponnese before finally settling in Thebes,
where the famous Theban general, Epaminondas, became his pupil and
called him father. In order to be the teacher of Epaminondas in the
early fourth century, Lysis must have been born no earlier than about
470. Thus the conflagration that he escaped as a young man must have
been part of the attacks on the Pythagoreans around 450, rather than
those that occurred around 500, when Pythagoras himself was still
alive. The later sources often conflate these two attacks on the
Pythagoreans (Minar 1942, 53). Nothing is known of the philosophy of
Lysis, but it seems probable that he should be regarded as one of the
acusmatici, since his training of Epaminondas appears to have
emphasized a way of life rather than mathematical or scientific studies
(Diodorus Siculus X 11.2) and Epaminondas' use of the name father
for Lysis suggests a cult association (Burkert 1972a, 179). In the
later tradition, Lysis became quite famous as the author of a spurious
letter (Thesleff 1965, 111; cf. Iamblichus, VP 75–78) rebuking
a certain Hipparchus for revealing Pythagorean teachings to the
uninitiated (see on the Pythagorean pseudepigrapha below, sect.
4.2).

Zopyrus of Tarentum is mentioned twice, in a treatise on
siege-engines by Biton (3rd or 2nd century BCE), as the inventor of an
advanced form of the type of artillery known as the belly-bow (Marsden
1971, 74–77). Zopyrus' bow used a winch to pull back the string
and hence could shoot a six-foot wooden missile 4.5 inches thick
(Marsden 1969, 14). It is not implausible to suppose that this is the
same Zopyrus as is listed in Iamblichus' catalogue of
Pythagoreans under Tarentum (Diels 1965, 23), although Biton does not
call him a Pythagorean. The traditional dating for Zopyrus puts him in
the first half of the fourth century (Marsden 1971, 98, n. 52), but
Kingsley has convincingly argued that he was in fact active in the last
quarter of the fifth century, when he designed artillery for Cumae and
Miletus (1995, 150 ff.). In a famous passage, Diodorus reports that in
399 BCE Dionysius I, the tyrant of Syracuse, gathered together skilled
craftsmen from Italy, Greece and Carthage in order to construct
artillery for his war with the Carthaginians (XIV 41.3). It seems not
unlikely that Zopyrus was one of those who came from Italy. There is no
reason to suppose, however, as Kingsley (1995, 146) and others do, that
Zopyrus' interest in mechanics was connected to his
Pythagoreanism or that there was a specifically Pythagorean school of
mechanics in Tarentum (Huffman 2005, 14–17).

It is controversial whether this Zopyrus of Tarentum is the same as
Zopyrus of Heraclea, who is not called a Pythagorean in the sources,
but who is reported in late sources to have written three Orphic poems,
The Net, The Robe and The Krater, which
probably dealt with the structure of human beings and the earth (West
1983, 10 ff.). This Zopyrus could be from the Heraclea closely
connected to Tarentum, but he might also be from the Heraclea on the
Black Sea. A late source connects Zopyrus of Heraclea with Pisistratus
in the 6th century (West 1983, 249), which would mean that he could not
be the same as Zopyrus of Tarentum in the late 5th century. On the
other hand, Orphic writings are assigned to a number of other
Pythagoreans, and it is not impossible that the same person had
interests both in Orphic mysticism and mechanics. Kingsley supposes
that the myth at the end of Plato's Phaedo is based in minute
detail on Zopyrus' Krater or an intermediary reworking
of it (1995, 79–171), and tries to connect specific features of the
myth to Zopyrus' interest in mechanics (1995, 147–148), but the
parallel which he detects between the oscillation of the rivers in the
mythic account of the underworld and the balance of opposing forces
used in a bow is too general to be compelling. The connection between
Zopyrus and the Phaedo is highly conjectural and must remain
so, as long as there are no fragments of the Krater, with
which to compare the Phaedo.

A harmonic theorist named Simus is accused of having plagiarized one
of seven pieces of wisdom inscribed on a bronze votive offering, which
was dedicated in the temple of Hera on Pythagoras' native island
of Samos, by Pythagoras' supposed son Arimnestus (Duris of Samos
in Porphyry, VP 3). There is a Simus listed under Posidonia
(Paestum in S. Italy) in Iamblichus' catalogue of Pythagoreans,
so that DK treated him as a Pythagorean (I 444–445) who, like Hippasus,
stole some of the master's teaching for his own glory. There is,
however, no obvious connection between the two individuals named Simus
except the name. Most scholars have thus treated Simus as if he were a
harmonic theorist in competition with and independent of the
Pythagorean tradition (Burkert 1972a, 449–450; Zhmud 2012a, 118; West
1992, 79 and 240; Wilamowitz 1962, II 93–94).

What exactly he stole is very unclear. He is said to have removed
seven pieces of wisdom from the monument and put forth one of them as
his own. This is perhaps best understood as meaning that he took an
inscribed piece of metal from the dedicated object, perhaps a cauldron
(see Wilamowitz 1962, II 94). The inscription will have included all
seven pieces of wisdom, but Simus chose to publish only one of them as
his own, the other six being thus lost. The piece of wisdom he put
forth as his own is called a kanôn (“rule”).
West takes this as a reference to the monochord, which was called the
kanôn, used to determine and illustrate the numerical
ratios, which were related to the concordant intervals (1992, 240).
Since, however, the kanôn seems to have been something
inscribed on the dedication, along with six other pieces of wisdom, it
is perhaps better to assume that the kanôn was a
description of a set of ratios determining a scale (Burkert 1972a,
455; Wilamowitz 1962, 94). There must have been a scale in circulation
associated with the name of Simus. The story that Duris reports is
then an attempt by the Pythagoreans to claim this scale as, in fact,
the work of Pythagoras or his son, which Simus plagiarized. Duris
wrote in the first part of the third century BCE, so Simus has to be
earlier than that. If the son of Pythagoras really made the dedication
in the temple, this would have occurred in the fifth century, but it
is unclear how much later than that Simus' kanôn became
known. West dates him to the fifth century, whereas DK places him in
the fourth. Zhmud suspects that he is an invention of the
pseudo–Pythagorean tradition (2012a, 118).

Iamblichus describes an ‘arithmetical method’ known as
the bloom of Thymaridas (In Nic. 62), and elsewhere discusses
two points of terminology in Thymaridas, including his definition of
the monad as “limiting quantity” (In Nic. 11 and
27). Some scholars have dated Thymaridas to the time of Plato or
before, but others argue that the terminology assigned to him cannot be
earlier than Plato and shows connections to Diophantus in the third
century CE (see Burkert 1972a, 442, n. 92 for a summary of the
scholarship). There is also a Thymaridas in the biographical tradition,
who may or may not be the same individual. In a highly suspect passage
in Iamblichus, Thymarides is listed as a pupil of Pythagoras himself
(VP 104) and a Thymaridas of Paros appears in
Iamblichus' catalogue and is mentioned in one anecdote
(VP 239). There is also a worrisome connection to the
pseudo-Pythagorean literature. A Thymaridas of Tarentum is presented in
an anecdote (Iamblichus, VP 145) as arguing that people should
wish for what the gods give them rather than praying that the gods give
them what they want, a sentiment that is also found in a group of three
treatises forged in Pythagoras' name (Diogenes Laertius VIII 9).
The anecdote is drawn from Androcydes' work on the Pythagorean
symbola or taboos. If this work could be dated to the fourth
century, it would confirm an early date for Thymaridas, but all that is
certain is that Androcydes' work was known in the first century
BCE and thus that the anecdote originated before that date (Burkert
1972a, 167). It seems rash, given this confused evidence, to follow
Zhmud and regard Thymaridas as a younger contemporary or pupil of
Archytas (2012a, 131).

Aristoxenus (ca. 375- ca. 300 BCE) is most famous as a music theorist
and as a member of the Lyceum, who was disappointed not be to named
Aristotle's successor (Fr. 1 Wehrli). In his early years, however, he
was a Pythagorean, and he is one of the most important sources for
early Pythagoreanism. He wrote five works on Pythagoreanism, although
it is possible that some of these titles are alternative names for the
same work: The Life of Pythagoras, On Pythagoras and His
Associates, On the Pythagorean Life,
Pythagorean Precepts and a Life of Archytas. None of
these works have survived intact, but portions of them were preserved
by later authors (Wehrli 1945). Aristoxenus is a valuable source
because, as a member of the Lyceum, he is free of the distorted image
of Pythagoras propagated during his lifetime by Plato's successors in
the Academy (see below, sect. 4.1) and because of his unique
connections to Pythagoreanism.

He was born in Tarentum during the years when the most important
Pythagorean of the fourth century, Archytas, was the leading public
figure and his father, Spintharus, had connections to Archytas (Fr. 30
Wehrli). When Aristoxenus left Tarentum, as a young man, and
eventually came to Athens (ca. 350), his first teacher was the
Pythagorean, Xenophilus, before he went on to become the pupil of
Aristotle (Fr. 1 Wehrli). Some modern scholars are skeptical of
Aristoxenus' testimony, seeing his denial that there was a prohibition
on eating beans and his assertion that Pythagoras was not a vegetarian
and particularly enjoyed eating young pigs and tender kids (Fr. 25 =
Gellius IV 11), as attempts to make Pythagoreanism more rational than
it was (Burkert 1972a, 107, 180). On the other hand, his Life of
Archytas is not a simple panegyric; Archytas' foibles are
recognized and his opponents are given a fair hearing (Huffman 2005,
4–5, 289–290). On Aristoxenus as a source for
Pythagoreanism see most recently Zhmud 2012b and Huffman 2014b,
285–295.

Perhaps Aristoxenus' most interesting work on Pythagoreanism is
the Pythagorean Precepts, which is known primarily through
substantial excerpts preserved by Stobaeus (Frs. 33–41 Wehrli). This
work does not mention any Pythagoreans by name but presents a set of
ethical precepts that “they” (i.e. the Pythagoreans)
proposed concerning the various stages of human life, education, and
the proper place of sexuality and reproduction in human life. There
are also analyses of concepts important in ethics, such as desire and
luck. Given Aristoxenus' background, the Precepts would
appear to be invaluable evidence for Pythagorean ethics in the first
half of the fourth century, when Aristoxenus was studying
Pythagoreanism. They might be expected to partially embody the views
of his teacher Xenophilus. The standard scholarly view of this work,
however, is that Aristoxenus plundered Platonic and Aristotelian ideas
for the glory of the Pythagoreans (Wehrli 1945, 58 ff.; Burkert 1972a,
107–108; Zhmud 2012a, 65). There are serious difficulties with the standard view,
however (Huffman 2008b). The analysis of luck that was supposedly
taken from Aristotle is, in fact, in sharp conflict with Aristotle's
view (Mills 1982) and appears to be one of the views Aristotle was
attacking. While the Precepts do have similarities to
passages in Plato and Aristotle, they are at a very high level of
generality and are shared with passages in other fifth and fourth
century authors, such as Xenophon and Thucydides; it is the
distinctively Platonic and Aristotelian features that are missing.

The Precepts are thus best regarded as what they appear on
the surface to be, an account of Pythagorean ethics of the fourth
century. This ethical system shows a similarity to a conservative
strain of Greek ethics, which is also found in Plato's
Republic, but has its own distinctive features (Huffman
2006). The central outlook of the Precepts is a distrust of
basic human nature and an emphasis on the necessity for supervision of
all aspects of human life (Fr. 35 Wehrli). The emphasis on order in
life is so marked that the status quo is preferred to what is
right (Fr. 34). The Pythagoreans were particularly suspicious of
bodily desire and analyzed the ways in which it could lead people
astray (Fr. 37). There are strict limitations on sexual desire and the
propagation of children (Fr. 39). Despite the best efforts of
humanity, however, many things are outside of human control, so the
Pythagoreans examined the impact of luck on human life (Fr. 41).

Aristoxenus is a source for the famous story of the two Pythagorean
friends Damon and Phintias, which was set during the tyranny of
Dionysius II in Syracuse (367–357). As a test of their friendship
Dionysius falsely accused Phintias of plotting against him and
sentenced him to death. Phintias asked time to set his affairs in
order, and Dionysius was amazed when Damon took his place, while he did
so. Phintias showed his equal devotion to his friend by showing up on
time for his execution. Dionysius cancelled the execution and asked to
become a partner in their friendship but was refused (Iamblichus,
VP 234; Porphyry, VP 59–60; Diodorus X 4.3).

In Diodorus' version, Phintias is presented as actually
engaged in a plot against Dionysius and some argue that
Aristoxenus' version is an attempt to whitewash the Pythagoreans
(Riedweg 2005, 40). On the other hand, Dionysius' eagerness to
join in their friendship, which occurs in both versions, is harder to
understand if there really had been a plot (see Burkert 1972a, 104).
There are two other considerations. First, Aristoxenus cites Dionysius
II himself as his source, whereas it is unclear what source Diodorus
used. Second, it is far from clear that Aristoxenus would object to the
Pythagoreans plotting against a tyrant. Thus, there are good reasons
for regarding Aristoxenus' version as more accurate.

Cleinias and Prorus are another pair of Pythagorean friends, whose
story may have been told by Aristoxenus (Iamblichus, VP 127),
although they were not friends in the usual sense. Cleinias, who was
from Tarentum, knew nothing of Prorus of Cyrene other than that he was
a Pythagorean, who had lost his fortune in political turmoil. On these
grounds alone he went to Cyrene, taking the money to restore
Prorus' fortunes (Iamblichus, VP 239; Diodorus X 4.1).
Nothing else is known of Prorus, although some pseudepigrapha were
forged in his name (Thesleff 1965, 154.13). It appears that Cleinias
was a contemporary of Plato, since Aristoxenus reports that he and an
otherwise unknown Pythagorean, Amyclas, persuaded Plato not to burn the
books of Democritus, on the grounds that it would do no good, since
they were already widely known (Diogenes Laertius IX 40). Cleinias was
involved in several other anecdotes. Like Archytas he supposedly
refused to punish when angry (VP 198) and, when angered,
calmed himself by playing the lyre (Athenaeus XIV 624a). Asked when one
should resort to a woman he said “when one happens to want
especially to be harmed” (Plutarch, Moralia 654b).
Several pseudepigrapha appear in Cleinias' name as well.

Myllias of Croton and his wife Timycha appear in Iamblichus'
catalogue and are known from a famous anecdote of uncertain origin,
which is preserved by Iamblichus (VP 189 ff.). They were
persecuted by the tyrant Dionysius II of Syracuse, but Timycha showed
her loyalty and courage by biting off her tongue and spitting it in the
tyrant's face, rather than risk divulging Pythagorean secrets under
torture.

None of the Pythagoreans mentioned in the previous four paragraphs
appear to have to have anything to do with the sciences or natural
philosophy. Since their Pythagoreanism consists exclusively in their
way of life, they are best regarded as examples of the
acusmatici. Many scholars have regarded Diodorus of Aspendus
in Pamphylia (southern Asia Minor), as an important example of what the
Pythagorean acusmatici were like in the first half of the
fourth century (Burkert 1972a, 202–204). Diodorus is primarily known
through a group of citations preserved by Athenaeus (IV 163c-f), which
describe him as a vegetarian who was outfitted in an outlandish way,
some features of which later became characteristic of the Cynics, e.g.,
long hair, long beard, a shabby cloak, a staff and beggar's rucksack
(cf. Diogenes Laertius VI 13). The historian Timaeus (350–260),
however, casts doubt on Diodorus' credentials as a Pythagorean
saying that “he pretended to have associated with the
Pythagoreans” and Sosicrates, another historian (2nd century BCE;
fragments in Jacoby) says that his outlandish dress was his own
innovation, since before this Pythagoreans had always worn white
clothing, bathed and wore their hair according to fashion (Athenaeus IV
163e ff.). Iamblichus, the other major source for Diodorus outside
Athenaeus, also treats Diodorus with reserve, saying that he was
accepted by the leader of the Pythagorean school at the time, one
Aresas, because there were so few members of the school. He
continues, perhaps again with disapproval, to report that Diodorus
returned to Greece and spread abroad the Pythagorean oral
teachings.

These sources clearly suggest that Diodorus was anything but a
typical Pythagorean, even of the acusmatic variety. Burkert
has argued that this reflects a bias of sources such as Aristoxenus,
who wanted to make Pythagoreanism appear reasonable and emphasized the
version of Pythagoreanism practiced by the mathêmatici
rather than the acusmatici. In support of this conclusion, he
argues that the two earliest sources present Diodorus as a Pythagorean
without any qualifications (1972a, 204). It is important to look
carefully at those sources, however. First, neither is a philosopher or
a historian, who might be expected to give a careful presentation of
Diodorus. The oldest is a lyre player named Stratonicus (died 350 BCE),
who was famous for his witticisms, and the other, Archestratus (fl. 330
BCE), wrote a book entitled The Life of Luxury, which focused
on culinary delights. Such sources might be expected to accept typical
stories that went around about Diodorus without any close analysis.

In the case of our earliest source, Stratonicus, there is, moreover,
once again evidence suggesting that Diodorus was not regarded as a
typical Pythagorean. In describing Diodorus' relationship to
Pythagoras, Stratonicus does not use a typical word for student or
disciple, but rather the same word (pelatês) that Plato
used in the Euthyphro to describe the day-laborer who died at
the hands of Euthyphro's father. Diodorus is thus being presented
sarcastically as a hired hand in the Pythagorean tradition, which is
very much in accord with the later presentations of him as a poor man's
Pythagoras on the fringes of Pythagoreanism. Thus, rather than accusing
the sources of bias against Diodorus, it seems better to accept their
almost universal testimony that he was not a typical acusmatic
but rather a marginal figure, who used Pythagoreanism in part to try to
gain respectability for his own eccentric lifestyle.

Individuals known as “Pythagorists,” i.e. Pythagorizers,
are ridiculed by writers of Greek comedy, such as Alexis, Antiphanes,
Aristophon, and Cratinus the younger, in the middle and second half of
the fourth century (see Burkert 1972a, 198, n. 25 for the evidence and
200, n. 41 for the dating). The most important of the fragments of
these comedies that deal with the Pythagorists are collected by
Athenaeus (IV 160f ff) and Diogenes Laertius (VIII 37–38). The term
“Pythagorist” is usually negative in the comic writers
(Arnott 1996, 581–582) and picks out people who share some of the same
extreme ascetic lifestyle as Diodorus. A fragment of Antiphanes
describes someone as eating “nothing animate, as if
Pythagorizing” (Fr. 133 Kassel and Austin = Athenaeus IV 161a).
In The Pythagorizing Woman, Alexis presents the vegetarian
sacrificial feast that is customary for the Pythagoreans as including
dried figs, cheese and olive cakes, and reports that the Pythagorean
life entailed “scanty food, filth, cold, silence, sullenness, and
no baths” as well as drinking water instead of wine (Frs. 201–202
= Athenaeus IV 161c and III 122f).

A number of these characteristics can be connected to the
acusmata (Arnott 1996, 583), e.g., the lack of bathing may be
a joke based on the acusma that forbids the Pythagoreans from
using the public baths (Iamblichus, VP 83), Antiphanes
(fr. 158) satirizes the acusmata's bizarre list of foods that
can be eaten (D.L. 8.19) by describing his Pythagoreans as searching
for sea orach, and the silence or sullenness ascribed to the
Pythagoreans in comedy accords not just with the acusmata but
with early testimony about the Pythagoreans in Isocrates
(Busiris 29) and Dicaearchus (Fr. 40 Mirhady). A fragment of
Aristophon's
Pythagorist suggests that this ascetic life was based on
poverty rather than philosophical scruple and that, if you put meat
and fish in front of these Pythagorists, they would gobble them down
(Fr. 9 = Athenaeus IV 161e). In a fragment of Alexis, after the
speaker reports that the Pythagoreans eat nothing animate, he is
interrupted by someone who objects that “Epicharides eats dogs,
and he is a Pythagorean,” to which the response is, “yes,
but he kills them first and so they are not still animate”
(Fr. 223 + Athenaeus 161b). Epicharides and some other named figures
may well be Athenians who are satirized by being assigned a
Pythagorean life (Athenaeus 2006, 272). Another fragment of
Aristophon's Pythagorist reports that the Pythagoreans have a
far different existence in the underworld than others, in that they
feast with Hades because of their piety, but this just occasions the
remark that Hades is an unpleasant god to enjoy the company of such
filthy wretches (Fr. 12 = Diogenes Laertius VIII 38).

Both Alexis (Fr. 223 = Athenaeus IV 161b) and Cratinus the younger
(Fr. 7 = Diogenes Laertius VIII 37) wrote plays entitled The People
of Tarentum, which, although they may not have been primarily
about Pythagoreans, featured depictions of them (Arnott 1996, 625–626).
In this case, the Pythagoreans are again satirized for their simple
diet, bread and water (which is called “prison fare”), and
for drinking no wine. In these plays, however, the Pythagoreans are
also presented as feeding on “subtle arguments” and
“finely honed thoughts” and as pestering others with them,
in a way that is reminiscent of Aristophanes' treatment of
Socrates in the Clouds.

Given the fragmentary nature of the evidence, it is unclear whether
these ascetic Pythagoreans who engage in argument are the same as the
Pythagorists in the other comedies, who are characterized by their
filth and eccentric appearance. Certainly the latter are more
reminiscent of Diodorus of Aspendus, while the former might be closer
to what we know of someone like Cleinias. In the first half of the
third century, the poet Theocritus still preserves a memory of these
Pythagorists as “pale and without shoes” (XIV 5). The
scholiast to the passage testifies to the continuing controversy about
the Pythagorists by drawing a distinction between Pythagoreans who give
every attention to their body and Pythagorists who are filthy (although
another scholion reports that others say the opposite, see Arnott 1996,
581). A passage in Iamblichus (VP 80) similarly argues that
the Pythagoreans were the true followers of Pythagoras, while the
Pythagorists just emulated them.

In recent scholarship, the tendency has been to regard Diodorus and
the Pythagorists as legitimate Pythagoreans of the acusmatic stamp,
whose eccentricities are perhaps a little exaggerated in comedy. The
extensive evidence from antiquity which argues that they were not true
Pythagoreans is interpreted as bias on the part of conservative
Pythagoreans of the hyper-mathêmatici sort, such as
Aristoxenus, who wanted to disassociate themselves and Pythagoreanism
in general from such strange people. This is a possible interpretation
of the evidence, but, as the evidence for Diodorus shows, it is also
quite possible that Diodorus and the more extreme Pythagorists depicted
in comedy were in fact people with whom few Pythagoreans either of the
mathêmatici or the acusmatici wanted to
associate themselves. Many religious movements have a radical fringe,
and there is little reason to think that Pythagoreanism should differ
in this regard. In connection with his thesis that the
acusmata were a literary phenomenon and that no one lived a life
in accordance with them Zhmud argues that the Pythagorists of comedy
are a creation of the comic stage and do not provide evidence for
Pythagoreans living a life governed by acusmata (Zhmud 2012a,
175-183). It is true that many of the features of the Pythagorists are
shared with Socrates as presented in the Clouds (subtle
arguments, plain food, filthy clothes). Zhmud suggests that
vegetarianism was added to this stock picture of the philosopher to
give a Pythagorean color and that this vegetarianism was derived
solely from the eccentric figure of Diodorus of Aspendus. However, as
noted above there are more connections to the acusmata than
just vegetarianism and it is hard to believe that the repeated jokes
at the expense of those living a Pythagorean life had no correlate in
reality other than Diodorus.

Perhaps the best way to evaluate the complicated evidence for
fourth-century Pythagoreanism is to conclude that there were three main
groups, each of which admitted some variation. There were
mathêmatici such as Archytas who did serious research in
the mathematical disciplines and natural philosophy but who also lived
an ascetic life that emphasized self-control and avoidance of bodily
pleasure. Other Pythagoreans such as Cleinias or Xenophilus may have
done no work in the sciences but lived a Pythagorean life, which was
similar to that of Archytas and followed principles similar to those
set out in Aristoxenus' Pythagorean Precepts. They may
have observed some mild dietary restrictions and may be similar to the
figures satirized in The Men of Tarentum as eating a simple
diet but still engaged in subtle arguments. There was probably a
continuum of people in this category with some following more or
different sets of the acusmata than others. Finally there are
the Pythagorean hippies such as Diodorus and the Pythagorists, who
ostentatiously live a life in accord with some of the
acusmata, but who take such an extreme interpretation of them
as to be regarded as eccentrics by most Pythagoreans.

Diogenes Laertius reports, evidently on the authority of
Aristoxenus, that the last Pythagoreans were Xenophilus from the
Thracian Chalcidice (Aristoxenus' teacher), and four Pythagoreans
from Phlius: Phanton, Echecrates, Diocles and Polymnastus. These
Pythagoreans are further identified as the pupils of Philolaus and
Eurytus. Little more is known of Xenophilus beyond his living for more
than 105 years (DK I 442–443). The Pythagoreans from Phlius are just
names except Echecrates (DK I 443), to whom Phaedo narrates, evidently
in Phlius, the events of Socrates' last day in Plato's Phaedo.
Socrates' interlocutors in the Phaedo, Simmias and
Cebes, are often regarded as Pythagoreans, because they are said to
have been pupils of Philolaus when he was in Thebes. They are also
shown to be pupils of Socrates, however, and it is unclear that their
connection to Philolaus was any closer than their connection to
Socrates. They are not listed in Iamblichus' catalogue as
Pythagoreans; Diogenes Laertius includes them with other followers of
Socrates (II 124–125). Echecrates might have been born around 420 and
thus be a young man at the dramatic date of the Phaedo.
Aristoxenus' assertion that these were the last of the
Pythagoreans would then suggest that Pythagoreanism died out around
350, when Echecrates was an old man.

Riedweg says that this claim is “demonstrably untrue”
pointing to a Pythagorean, Lycon, who criticized Aristotle's supposed
extravagant way of life and to the Pythagorists discussed above (2005,
106). This seems slender evidence upon which to be so critical of
Aristoxenus. Virtually nothing is known of Lycon, and Aristocles
(1st-2nd c. CE), who recounts the criticism of Aristotle, says that
Lycon “called himself a Pythagorean,” thus expressing some
sort of reservation about his credentials (DK I 445–446).
Aristoxenus' assertion is probably to be understood as a general claim
that, with the deaths of the Pythagoreans from Phlius around the
middle of the fourth century, Pythagoreanism as an active movement was
dead. This would be compatible with a few individuals still claiming
to be Pythagoreans after 350.

This is not inconsistent with the existence of a few isolated
individuals, who still claim to be Pythagoreans. Certainly, from the
evidence available to modern scholars, Aristoxenus' claim is
largely true. From about 350 BCE until about 100 BCE, there is a radical
drop in evidence for individuals who call themselves Pythagoreans.
Iamblichus (In Nic. 116.1–7) appears to date the Pythagoreans
Myonides and Euphranor, who worked on the mathematics of means, after
the time of Eratosthenes (285–194 BCE) and hence to the second century
BCE or later (Burkert 1972a, 442), but Iamblichus' history of the
means is very confused and they might belong to the rise of
Neopythagoreanism in the first centuries BCE and CE. Kahn (2001, 83)
sees a hint of Pythagorean cult activity in the spurious
Pythagorean Memoirs, which must date sometime before the
first half of the first century BCE, when they are quoted by Alexander
Polyhistor (see section 4.2 below). A few other Pythagorean
pseudepigrapha appear in the period (see further below, sect. 4.2),
although it is unclear what sort of Pythagorean community, if any, was
associated with them. Pythagoreanism is not completely dead between
350 and 100 (see further below, sect. 3.5), but few individual
Pythagoreans or organized groups of Pythagoreans can be identified in
this period.

The names Timaeus of Locri and Ocellus of Lucania are famous as the
authors of the two most influential Pythagorean pseudepigrapha (see
below, sect. 4.2). In his catalogue of Pythagoreans, Iamblichus lists
an Ocellus under Lucania and two men named Timaeus, neither under
Locri. The later forgery of works attributed to Timaeus and Ocellus
does not of course mean that Pythagoreans of these names did not exist,
and it is possible that the Timaeus of Locri who is the main speaker in
Plato's Timaeus was an historical Timaeus (some have thought
Plato uses him as a mask for Archytas, however). If they really did
exist, however, nothing is known about them, since all other reports in
the ancient tradition are likely to be based on Plato's
Timaeus or the spurious works in their name.

Some scholars have argued that Hicetas and Ecphantus, both of
Syracuse, were not historical figures at all but rather characters in
dialogues written by Heraclides of Pontus, a fourth-century member of
the Academy. By a misunderstanding, they came to be treated as
historical Pythagoreans in the doxographical tradition (see Guthrie
1962, 323 ff. for references). This theory arose because both Hicetas
and Ecphantus are said to have made the earth rotate on its axis, while
the heavens remained fixed, in order to explain astronomical phenomena,
and, in one report, Heraclides is paired with Ecphantus as having
adopted this view (Aetius III 13.3 =DK I 442.23). In addition Ecphantus
is assigned a form of atomism (DK I 442.7 ff.) similar to that assigned
to Heraclides (Fr. 118–121 Wehrli). It is not uncommon in the
doxographical tradition for a report of the form “x and y believe
z” to mean that “y, as reported by x, believes z,” so
it is suggested that in this case “Heraclides and
Ecphantus” means “Ecphantus as presented by
Heraclides.” There is a serious problem with this ingenious
theory. The doxographical reports about Hicetas and Ecphantus
ultimately rely on Theophrastus (Cicero mentions Theophrastus by name
at DK I 441.27), and it is implausible that Theophrastus would treat
characters invented by his older contemporary, Heraclides, as
historical figures. Theophrastus did accept the Academic glorification
of Pythagoras (see on Neopythagoreanism below, sect. 4.1), but this
provides no grounds for supposing that he accepted a character in a
dialogue as a historical person (pace Burkert 1972a, 341).

The testimonia for Hicetas are meager and contradictory (DK I
441–442). He appears to have argued that the celestial phenomena are
best explained by assuming that all heavenly bodies are stationary and
that the apparent movement of the stars and planets is the result of
the earth's rotation around its own axis. He may also have followed
Philolaus in positing a counter-earth, opposite the earth on the other
side of a central fire, although, if he did, it is unclear how he
would have explained why it and the central fire are not visible from
the rotating earth. In Philolaus' system the central fire remains
invisible because the earth orbits the central fire as it rotates on
its axis, thus keeping one side of the earth always turned away from
the central fire. A little more is known about Ecphantus (DK I
442). He too is said to have believed that the earth moved, not by
changing its location (as Philolaus proposed, in making the earth and
counter-earth revolve around the central fire: see Section 4.2 of the
entry on
Philolaus),
but by rotating on its axis.

Copernicus was inspired by these testimonia about Hicetas and
Ecphantus, as well as those about Philolaus, to consider the motion of
the earth (see below, sect. 5.2). Ecphantus developed his own original
form of atomism. He is best understood as reacting to and developing
the views of Democritus. He agreed with Democritus 1) “that human
beings do not grasp true knowledge of the things that are, but define
them as they believe them to be” (DK I 442.7–8; cf. Democritus
Frs. 6–10) and 2) that all sensible things arise from indivisible first
bodies and void. He differs from Democritus, however, in supposing that
atoms are limited rather than unlimited in number and that there is
just one cosmos rather than many. As in Democritus, atoms differ in
shape and size, but Ecphantus adds power (dynamis) as a third
distinguishing factor. He explains atomic motion not just in terms of
weight and external blows, as the atomists did, but also by a divine
power, which he called mind or soul, so that “the cosmos was
composed of atoms but organized by providence” (DK I 442.21–22).
It is because of this divine power that the cosmos is spherical in
shape. This unique spherical cosmos is reminiscent of Plato's
Timaeus, but the rest of Ecphantus' system differs
enough from Plato that there is no question of its being a forgery
based on the Timaeus. One testimony says that he was the first
to make Pythagorean monads corporeal, thus differing from the
fifth-century Pythagoreans described by Aristotle, who do not seem to
have addressed the question of whether numbers were physical entities
or not (Huffman 1993, 61 ff.).

It is difficult to be sure of the date of either Hicetas or
Ecphantus. Since, however, both seem to be influenced by Philolaus'
idea of a moving earth and since Ecphantus appears to be developing
the atomism of Democritus, it is usually assumed that they belong to
the first half of the fourth century (Guthrie 1962, 325–329;
Zhmud 2012a, 130). Hicetas does not appear in Iamblichus'
catalogue. There is an Ecphantus in the catalogue, but he is listed
under Croton rather than Syracuse, so it cannot be certain whether he
is the Ecphantus described in the doxography.

There is currently a very wide range of opinions about the
relationship of Plato to Pythagoreanism. Many scholars both ancient
and modern have thought that Plato was very closely tied to
Pythagoreanism. In the biography of Pythagoras read by Photius in the
9th century CE (Bibl. 249) Plato is presented as a member of
the Pythagorean school. He is the pupil of Archytas and the ninth
successor to Pythagoras himself. If this were true then Plato would
certainly be the most illustrious early Pythagorean after Pythagoras
himself. Some modern scholars, while not going this far, have seen the
connections between Plato and the Pythagoreans to be very close
indeed. Thus, A. E. Taylor in his great commentary on
the Timaeus says that his main thesis is that “the
teaching of Timaeus [in Plato's Timaeus] can be shown to be
in detail exactly what we should expect from an fifth-century Italian
Pythagorean” (1928, 11). Guthrie in his famous history of
ancient philosophy commented that Pythagorean and Platonic philosophy
were so close that it is difficult to separate them (1975,
35). Recently it has been argued that Plato was so steeped in
Pythagoreanism that he structured his dialogues by counting numbers of
lines and placing important passages at points in the dialogue that
correspond to important ratios in Pythagorean harmonic theory
(Kennedy, 2010 and 2011). Thus, the vision of the form of beauty
appears 3/4 of the way through the Symposium by line count
and the ratio 3 : 4 corresponds to the central musical interval of the
fourth. There are, however, serious questions about the methodology
used (Gregory 2012) and it is a serious problem both that no one in
the ancient world reports that Plato used such a practice and that the
middle of the dialogue, which corresponds to the most concordant
musical interval, the octave (2:1), does not usually contain the most
philosophically important content. Another approach sees Plato as
engaged with and heavily influenced by Pythagorean ideas in passages
where the Pythagoreans are not specifically mentioned in dialogues
such as the Cratylus (401b11-d7) and Phaedo
(101b10-104c9)(Horky 2013). The problem is that in contrast to
the Philebus, where the connection to Philolaus is clear (see
below), the connections to the Pythagoreans in these passages are too
indirect or general (e.g., the concepts odd and even and the number 3
in the Phaedo passage are not unique to the Pythagoreans) to
be very convincing and partly depend on the doubtful assumption that
Epicharmus was a Pythagorean (see section 3.4 above). The central text
for many of those who see Plato as closely tied to Pythagoreanism is
Aristotle's comment in Metaphysics 1.6 that Plato
“followed these men (i.e. the Pythagoreans according to these
scholars) in most respects” (987a29-31). In contrast to these
attempts to connect Plato closely to Pythagoreanism, most recent
Platonic scholars seem to think Pythagoreanism of little importance
for Plato. Thus two prominent handbooks to Plato's thought (Kraut
1992; Benson 2006) and another book of essays devoted specifically to
the Timaeus, (Mohr and Sattler 2010) hardly mention the
Pythagoreans at all.

In recent studies of the topic that lie somewhere between these
extremes, one approach is to argue that there is clear Pythagorean
influence on Plato but that its scope is much more limited than often
assumed (Huffman 2013b). Plato explicitly mentions Pythagoras and the
Pythagoreans only one time each in the dialogues and this
provides prima facie evidence that Pythagorean influence was
not extensive. Moreover, at Metaphysics 987a29-31 the
“these men” that Aristole says Plato follows in most
respects may not be the Pythagoreans but the Presocratics in
general. Aristotle's presentation as a whole mainly attests to
Pythagorean influence only on Plato's late theory of principles
(Huffman 2008a). It is often assumed that Plato owes his mathematical
conception of the cosmos and his belief in the immortality and
transmigration of the soul to Pythagoreanism (Kahn 2001,
3-4). However, the role of Pythagoreanism in Greek mathematics has
been overstated and while Plato had contacts with mathematicians who
were Pythagoreans like Archytas, the most prominent mathematicians in
the dialogues, Theodorus and Theaetetus, are not Pythagoreans. It is
thus a serious mistake to assume that any mention of mathematics in
Plato suggests Pythagorean influence. The same is true of the
immortality and transmigration of the soul in Plato, which are often
assumed to be derived from Pythagoreanism. Some have also thought that
Platonic myths and especially the myth at the end of
the Phaedo draw heavily on Pythagoreanism (Kingsley 1995,
79-171). However, most of the contexts in which Plato mentions the
immortality of the soul including the Platonic myths, suggest that he
is thinking of mystery cults and the Orphics rather than the
Pythagoreans (Huffman 2013b, 243-254). On the other hand, in
the Philebus (16c-17a) Plato gives clear acknowledgement of
the debt he owes to men before his time who posit limit and unlimited
as basic principles. The fragments of Philolaus and Aristotle's
reports on Pythagoreanism make clear that this is a reference to
Philolaus and the Pythagoreans. The principles of limit and unlimited
are clearly connected to Plato's one and indefinite dyad and it is
precisely these principles of Plato that Aristotle connects most
closely to Pythagoreanism (Metaph. 987b25-32). Thus Plato's
evidence coheres with Aristotle's to suggest that Pythagoreanism
exerted considerable influence on Plato's late theory of
principles. It is also true that specific aspects of Plato's
mathematical view of the world are owed to the Pythagoreans, e.g., the
world soul in the Timaeus is constructed according to the
diatonic scale that is prominent in Philolaus (Fr. 6a). However, most
of the Timaeus is not derived from Pythagoreanism and some of
it in fact conflicits with Pythagoreanism (e.g., Archytas famously
argued that the universe was unlimited while Plato's in limited). The
same is true for Plato as a whole. Isolated ideas such as the one and
the dyad and the structure of the world soul show heavy Pythagorean
influence, but there is no evidence that Pythagoreanism played a
central role in the development of the core of Plato's philosophy
(e.g., the theory of forms).

A second approach is to argue that, while it is true that not all
mentions of mathematics or all mentions of the transmigration of the
soul derive from Pythagoreanism, nonetheless a central system of value
that appears early in Plato's work and persists to the end is derived
from Pythagoreanism (Palmer 2014). Already in the Gorgias
Plato argues that principles of order and correctness which are found
in the cosmos and explain its goodness also govern human
relations. Socrates here puts forth a much more definite conception of
the good than in earlier dialogues. His complaint that Callicles pays
no attention to the role played by orderliness and self-control and
neglects geometrical equality (507e6-508a8) mirrors the emphasis on
organization and calculation in contemporary Pythagorean texts such as
Archytas Fr. 3 and Aristoxenus' Pythagorean Precepts
Fr. 35. It thus appears that “Socrates'” new insight into
the good in Gorgias derives from Plato's contact with the
Pythagoreans after the death of the historical Socrates. Plato never
abandons this Pythagorean conception of value and it can be traced
through the Phaedo and Republic to late dialogues
such as the Timaeus, where the cosmos is embued with
principles of mathematical order, and Philebus, where the
highest value is assigned to measure (66a). The question is whether
this emphasis on measure and order is uniquely Pythagorean in
origin.

Neopythagoreanism is characterized by the tendency to see Pythagoras
as the central and original figure in the development of Greek
philosophy, to whom, according to some authors (e.g. Iamblichus,
VP 1), a divine revelation had been given. This revelation
was often seen as having close affinities to the wisdom of earlier
non-Greeks such as the Hebrews, the Magi and the Egyptians. Because of
the belief in the centrality of the philosophy of Pythagoras, later
philosophy was regarded as simply an elaboration of the revelation
expounded by Pythagoras; it thus became the fashion to father the
views of later philosophers, particularly Plato, back onto
Pythagoras. Neopythagoreans typically emphasize the role of number in
the cosmos and treat the One and Indefinite Dyad as ultimate
principles going back to Pythagoras, although these principles in fact
originate with Plato. The origins of Neopythagoreanism are probably to
be found already in Plato's school, the Academy, in the second half of
the fourth century BCE. There is evidence that Plato's successors,
Speusippus and Xenocrates, both presented Academic speculations
arising in part from Plato's later metaphysics as the work of
Pythagoras, who lived some 150 years earlier. After a decline in
interest in Pythagoreanism for a couple of centuries,
Neopythagoreanism emerged again and developed further starting in the
first century BCE and extending throughout the rest of antiquity and
into the middle ages and Renaissance. During this entire period, it is
the Neopythagorean construct of Pythagoras that dominates, a construct
that has only limited contact with early Pythagoreanism; there is
little interest in an historically accurate presentation of Pythagoras
and his philosophy. In reading the following account of
Neopythagoreanism, it may be helpful to refer to the
Chronological Chart of Sources for Pythagoras,
in the entry on Pythagoras.

The evidence for Speusippus, Plato's successor as head of the Academy,
is fragmentary and second hand, so that certainty in interpretation is
hardly possible. In one passage, however, he assigns not just Plato's
principles, the one and the dyad, to “the ancients,” who
in context seem likely to be the Pythagoreans, but also a development
of the Platonic system according to which the one was regarded as
beyond being (Fr. 48 Taran; see Burkert 1972a, 63–64; Dillon
2003, 56–57). Some scholars reject this widely held view on the
grounds that this fragment of Speusippus is spurious (Zhmud 2012a,
424—425, who cites other scholars; Taran 1981, 350ff.; for a
response see Dillon 2014, 251) and if this were true it would
seriously weaken the case for supposing that Neopythagoreanism began
already in the Academy. Speusippus also wrote a book On
Pythagorean Numbers (Fr. 28 Taran), which builds on ideas
attested for the early Pythagoreans (e.g., ten as the perfect number,
although Zhmud regards the perfection of ten as a Platonic rather than
a Pythagorean doctrine 2012a, 404–09). We cannot be sure,
however, either that the title goes back to Speusippus or that he
assigned all ideas in it to the Pythagoreans. Aristotle twice cites
agreement between Speusippus and the Pythagoreans
(Metaph. 1072b30 ff.; EN 1096b5–8), which
might suggest that Speusippus himself had identified the Pythagoreans
as his predecessors in these areas. Speusippus and Xenocrates denied
that the creation of the universe in Plato's Timaeus should
be understood literally; when the view that the cosmos was only
created in thought and not in time is assigned to Pythagoras in the
later doxography (Aëtius II 4.1 — Diels 1958, 330), it
certainly looks as if an idea which had its origin in the
interpretation of Plato's
Timaeus in the Academy is being assigned back to Pythagoras
(Burkert 1972a, 71). The evidence is not sufficient to conclude that
Speusippus routinely assigned Platonic and Academic ideas to the
Pythagoreans (Taran 1981, 109), but there is enough evidence to suggest
that he did so in some cases.

Speusippus' successor as head of the Academy, Xenocrates, may actually
have followed some version of the Pythagorean way of life, e.g., he
was apparently a vegetarian, refused to give oaths, was protective of
animals and followed a highly structured daily regimen, setting aside
time for silence (Dillon 2003, 94–95 and 2014, 254–257;
Burkert, however, argues that he rejected metempsychosis [1972a,
124]). He wrote a book entitled Things Pythagorean, the
contents of which are unfortunately unknown (Diogenes Laertius IV
13). In the extant fragments of his writings, he refers to Pythagoras
by name once, reporting that “he discovered that the musical
intervals too did not arise apart from number” (Fr. 9
Heinze). Several doctrines of Xenocrates are also assigned to
Pythagoras in the doxographical tradition, e.g., the definition of the
soul as “a number moving itself,” which Xenocrates clearly
developed on the basis of Plato's
Timaeus (Plutarch, On the Generation of the Soul
1012d; Aëtius IV 2.3–4). This suggests that Xenocrates,
like Speusippus, may have assigned his own teachings back to
Pythagoras or at least treated Pythagoras as his precursor in such as
way that it was easy for others to do so (Burkert 1972a, 64–65;
Dillon 2003, 153–154; Zhmud [2012a, 55 and 426–427]
disputes this interpretation).

Yet another member of the early Academy, Heraclides of Pontus
(Gottschalk 1980), in a series of influential dialogues, further
developed the presentation of Pythagoras as the founder of philosophy.
In the dialogue, On the Woman Who Stopped Breathing,
Pythagoras is presented as the inventor of the word
“philosophy” (Frs. 87–88 Wehrli = Diogenes Laertius
Proem 12 and Cicero, Tusc. V 3.8). Although some scholars
have tried to find a kernel of truth in the story (e.g., Riedweg 2005,
90 ff., for a response see Huffman 2008c), its definition of the
philosopher as one who seeks wisdom rather than possessing it is
regarded by many scholars as a Socratic/Platonic formulation, which
Heraclides, in his dialogue, is assigning to Pythagoras as part of a
literary fiction (Burkert 1960 and 1972a, 65). Heraclides also
assigns to Pythagoras a definition of happiness as “the
knowledge of the perfection of the numbers of the soul” (Fr. 44
Wehrli), in which again the Platonic account of the numerical
structure of the soul in the Timaeus appears to be fathered
on Pythagoras. Other fragments show Heraclides' further fascination
with the Pythagoreans. He developed what would become one of the
canonical accounts of Pythagoras' previous incarnations (Fr. 89
Wehrli). Perhaps on the basis of the Pythagorean Philolaus'
astronomical system, he developed the astronomical theory, later to be
championed by Copernicus, according to which the apparent daily motion
of the sun and stars was to be explained by the rotation of the earth
(Frs. 104–108; see on Hicetas and Ecphantus above,
sect. 3.6). For a different view of Heraclides' relation to the
Pythagoreans see Zhmud 2012a, 427–432.

In contrast to the fascination with and glorification of Pythagoras in
the Academy after Plato's death, Aristotle did not treat Pythagoras as
part of the philosophical tradition at all. In the surveys of his
predecessors in his extant works, Aristotle does not include
Pythagoras himself and he evidently presented him in his lost special
treatises on the Pythagoreans only as a wonder-worker and founder of a
way of life. While Aristotle did acknowledge close connections
between Plato's late theory of principles (One and Indefinite Dyad)
and fifth-century Pythagoreans, he also sharply distinguished Plato
from the Pythagoreans on a series of important points
(Metaph. 987b23 ff.), perhaps in response to the Academy's
tendency to assign Platonic doctrines to Pythagoras. Aristotle's
students Eudemus, in his histories of arithmetic, geometry and
astronomy and Meno, in his history of medicine, follow Aristotle's
practice of not mentioning Pythagoras himself, referring to individual
Pythagoreans such as Philolaus or to the Pythagoreans as a
group. Eudemus assigns the Pythagoreans a number of important
contributions to the sciences but does not give them the decisive or
foundational role found in the Neopythagorean tradition. Aristotle's
pupils Dicaearchus (Porphyry, VP 19) and Aristoxenus do
mention Pythagoras but this is because they are focusing on the the
Pythagorean way of life and the history of the Pythagorean
communities. Neither assign to Pythagoras or the Pythagoreans the
characteristics of Neopythagoreanism. Aristoxenus is one of the most
important and extensive sources for Pythagoreanism (see 3.5 above). He
presents Pythagoras and the Pythagoreans in a positive manner but
avoids the hagiography and extravagant claims of the later
Neopythagorean tradition (Huffman 2014b). The standard view is that he
tries to emphasize the rational as opposed to the religious side of
Pythagoras (e.g. Burkert 1972a, 200–205), but several fragments
do highlight the religious aspect of Pythagoras' work (Huffman 2012b),
assigning him the doctrine of metempsychosis (fr. 12) and associating
him with the Chaldaean Zaratas (Fr. 13) and the Delphic oracle
(Fr. 15). It is only by rejecting the authenticity of such fragments
(as does Zhmud 2012a, 88–91) that Aristoxenus' account is purged
of religious elements. Dicaearchus' account of Pythagoreas is also
usually viewed as positive. He is supposed to have presented
Pythagoras as the model of the practical life as opposed to the
contemplative life (Jaeger 1948, 456; Kahn 2001, 68). However,
Dicaearchus presents a very sarcastic account of Pythagoras' rebirths
according to which he was reborn as the beautiful prostitute Alco
(Fr. 42) and careful reading of his other accounts of Pythagoras
suggests that he may have presented him as a charismatic charlatan who
bewitched his hearers (Fr. 42) and was seen as a threat to the
established laws of the state and hence was refused entrance by such
city-states as Locri (Fr. 41a. Huffman 2014b). Thus, Aristoxenus and
Dicaearchus were as divided in their interpretation of Pythagoras as
were Heraclitus and Empedocles in earlier centuries. The Peripatetic
tradition as a whole is in strong contrast, then, with the Academy
insofar as it emphasizes Pythagoreans rather than Pythagoras
himself. When Pythagoras is mentioned, it is mostly in connection with
the way of life, and interpretations range from positive to strongly
satirical but in either case avoid the hagiography of the
Neopythagorean tradition.

It is then one of the great paradoxes of the ancient Pythagorean
tradition that Aristotle's successor, Theophrastus, evidently accepted
the Academic lionization of Pythagoras, and identifies Plato's one and
the indefinite dyad as belonging to the Pythagoreans (Metaph.
11a27 ff.), although Aristotle is emphatic that this pair of
principles in fact belong to Plato
(Metaph. 987b25–27). Since Theophrastus'
work, Tenets in Natural Philosophy, was the basis of the
later doxographical tradition, it may be that Theophrastus is
responsible for the Neopythagorean Pythagoras of the Academy
dominating the later doxography, the Pythagoras who originated the one
and the indefinite dyad (Aëtius I 3. 8), but it may also be that
the Pythagorean sections of the doxography were rewritten in the first
century BCE, under the influence of the Neopythagoreanism of that
period (Burkert 1972a, 62; Zhmud 2012a, 455).

The standard view has thus been that the Academy was the origin of
Neopythagoreanism with its glorification of Pythagoras and its
tendency to assign mature Platonic views back to Pythagoras and the
Pythagoreans. Aristotle and the Peripatetics on the other hand
diminish the role of Pythagoras himself and, while noting connections
between Plato and the Pythagoreans, carefully distinguish Pythagorean
philosphy from Platonism. Zhmud has recently put forth a challenge to
this view arguing the the situation is almost the reverse: the Academy
in general regards Pythagoras and Pythagoreans favorably but does not
assign mature Platonic views to them, it is rather Aristotle who ties
Plato closely to the Pythagoreans (2012a, 415–456).

Although the origins of Neopythagoreanism are thus found in the
fourth century BCE, the figures more typically labeled Neopythagoreans
belong to the upsurge in interest in Pythagoreanism that begins in the
first century BCE and continues through the rest of antiquity. Before
turning to these Neopythagoreans, it is important to discuss another
aspect of the later Pythagorean tradition, the Pythagorean
pseudepigrapha. Many more writings forged in the name of Pythagoras and
other Pythagoreans have survived than genuine writings. Most of the
pseudepigrapha themselves only survive in excerpts quoted by
anthologists such as John of Stobi, who created a collection of Greek
texts for the edification of his son in early fifth century CE. The
modern edition of these Pythagorean pseudepigrapha by Thesleff (1965)
runs to some 245 pages.

There is much uncertainly as to when, where, why and by whom these
works were created. No one answer to these questions will fit all of
the treatises. Most scholars (e.g., Burkert 1972b, 40–44;
Centrone 1990, 30–34, 41–44 and 1994) have chosen Rome or
Alexandria between 150 BCE and 100 CE as the most likely time and
place for these compositions, since there was a strong resurgence of
interest in Pythagoreanism in these places at these times (see
below). Thesleff's view that the majority were composed in the third
century BCE in southern Italy (1961 and 1972, 59) has found less
favor. Centrone argues convincingly that a central core of the
pseudepigrapha were forged in the first centuries BCE and CE in
Alexandria, because of their close connection to Eudorus and Philo,
who worked in Alexandria in that period (Centrone 2014a). For an
overview of the Pythagorean pseudepigrapha see Centrone 2014a and
Moraux 1984, 605–683.

A number of motives probably led to the forgeries. The existence of
avid collectors of Pythagorean books such as Juba, King of Mauretania
(see below), and the scarcity of authentic Pythagorean texts will have
led to forgeries to sell for profit to the collectors. Other short
letters or treatises may have originated as exercises for students in
the rhetorical schools (e.g., the assignment might have been to write
the letter that Archytas wrote to Dionysius II of Syracuse asking that
Plato be freed; see Diogenes Laertius III 21–22). The contents
of the treatises suggest, however, that the primary motivation was to
provide the Pythagorean texts to support the Neopythagorean position,
first adumbrated in the early Academy, that Pythagoras was the source
of all that is true in the Greek philosophical tradition. The
pseudepigrapha show the Pythagoreans anticipating the most
characteristic ideas of Plato and Aristotle. Most of the treatises are
composed in the Doric dialect (spoken in Greek S. Italy) but, apart
from that concession to verisimilitude, there is little other attempt
to make them appear to be archaic documents that anticipated Plato and
Aristotle. Instead, Plato's and Aristotle's philosophical positions
are stated in a bald fashion using the exact Platonic and Aristotelian
terminology. In many cases, however, this glorification of Pythagoras
may not have been the final goal. The ancient authority of Pythagoras
was sometimes used to argue for a specific interpretation of Plato,
often an interpretation that showed Plato as having anticipated and
having responded to criticisms of Aristotle. For example, in defense
of the interpretation of Plato's Timaeus, which defends Plato
against Aristotle's criticisms by claiming that the creation of the
world in the Timaeus is metaphorical, a Platonist could point
to the forged treatise of Timaeus of Locri which does present the
generation as metaphorical but which can also be regarded as Plato's
source. These pseudo-Pythagorean treatises are adopting the same
strategy as Eudorus of Alexandria and thus may be more important for
debates within later Platonism than for Pythagoreanism per se
(Bonazzi 2013).

One plausible explanation of the sudden proliferation of Pythagorean
pseudepigrapha in the first century BCE and first century CE is the
reappearance of Aristotle's esoteric writings in the middle of the
first century BCE (Kalligas 2004, 39–42). In those treatises Plato is
presented as adopting a pair of principles, the one and the indefinite
dyad, which are not obvious in the dialogues, but which Aristotle
compares to the Pythagorean principles limit and unlimited
(e.g., Metaph. 987b19–988a1). Aristotle can be read, although
probably incorrectly (Huffman 2008a), as virtually identifying
Platonism and Pythagoreanism in these passages. Thus, Pythagorean
enthusiasts may have felt emboldened by this reading of Aristotle to
create the supposed original texts upon which Plato drew. They may
also have found support for this in Plato's making the south-Italian
Timaeus his spokesman in the dialogue of the same name. It is thus not
surprising that the most famous of the pseudepigrapha is the treatise
supposedly written by this Timaeus of Locri (Marg 1972), which has
survived complete and which is clearly intended to represent the
original document on which Plato drew, although it, in fact, also
responds to criticisms made of Plato's dialogue in the first couple of
centuries after it was written (Ryle 1965, 176–178). The treatise of
Timaeus of Locri is first mentioned by Nicomachus in the second
century CE (Handbook 11) and is thus commonly dated to the
first century CE. Another complete short treatise (13 pages in
Thesleff) is On the Nature of the Universe supposedly by the
Pythagorean Ocellus (Harder 1966), which has passages that are almost
identical to passages in Aristotle's On Generation and
Corruption. Since Ocellus' work is first mentioned by the Roman
polymath, Varro, scholars have dated it to the first half of the first
century BCE. Although Plato was in general more closely associated with
the Pythagorean tradition than Aristotle, a significant number of
Pythagorean pseudepigrapha follow ‘Ocellus’ in drawing on
Aristotle (see Karamanolis 2006, 133–135).

It is likely that in some cases letters were forged in order to
authenticate these forged treatises. Thus a correspondence between
Plato and Archytas dealing with the acquisition of the writings of
Ocellus (Diogenes Laertius VIII 80–81) may be intended to
validate the forgery in Ocellus' name (Harder 1966, 39ff). A letter
from Lysis to Hipparchus (Thesleff 1965, 111–114), which enjoyed
considerable fame in the later tradition and is quoted by Copernicus,
urges that the master's doctrines not be presented in public to the
uninitiated and recounts Pythagoras' daughter's preservation of his
“notebooks” (hypomnêmata) in secrecy,
although she could have sold them for much money (see Riedweg 2005,
120–121). Burkert (1961, 17–28) has argued that this
letter was forged to authenticate the “Pythagorean Notes”
from which Alexander Polyhistor (1st century BCE) derived his
influential account of Pythagoreanism (Diogenes Laertius VIII
24–36 — see the end of this section and for Alexander see
section 4.5 below). While some of Pythagoras' teachings were
undoubtedly secret, many were not, and the claim of secrecy in the
letter of Lysis is used to explain both the previous lack of early
Pythagorean documents and the recent “discovery” of what
are in reality forged documents, such as the notebooks.

There are fewer forged treatises in Pythagoras' name than in the name
of other Pythagoreans and they are a very varied group suggesting
different origins. Callimachus, in the third century BCE, knew of a
spurious astronomical work circulating in Pythagoras' name (Diogenes
Laertius IX 23) and there may have been a similar work forged in the
second century (Burkert 1961, 28–42). A group of three
books, On Education, On Statesmanship and On
Nature, were forged in Pythagoras' name sometime before the
second century BCE (Diogenes Laertius VIII 6 and 9; Burkert 1972a,
225). Heraclides Lembus, in the second century BCE, knew of at least
six other works in Pythagoras' name, all of which must have been
spurious, including a Sacred Discourse (Diogenes Laertius
VIII 7). The thesis that the historical Pythagoras wrote a Sacred
Discourse should be rejected (Burkert 1972a, 219). There was also
a spurious treatise on the magical properties of plants and the
Golden Verses, which are discussed further below (sect.
4.5). On the spurious treatises assigned to Pythagoras see Centrone 2014a, 316–318.

Archytas
appears to have been the most popular name in which to forge
treatises. Some 45 pages are devoted to
pseudo-Archytan treatises in Thesleff's collection as compared to 30
pages for Pythagoras (see Huffman 2005, 595 ff. for the
pseudo-Archytan treatises). The most famous of the pseudo-Archytan
texts is The Whole System of Categories, which, along
with On Opposites, represents the attempt to claim
Aristotle's system of categories for the Pythagoreans. The
pseudo-Archytan works on categories are very frequently cited by the
commentators on Aristotle's Categories (e.g., Simplicius and
Syrianus) and were regarded as authentic by them, but in fact include
modifications made to Aristotle's theory in the first century BCE and
probably were composed in that century (Szlezak 1972). Another
treatise, On Principles, is full of Aristotelian terminology
such as “form,” “substance,” and “what
underlies”; On Intelligence and Perception contains a
paraphrase of the divided line passage in
Plato's Republic. There are also a series of pseudepigrapha
on ethics by Archytas and other authors (Centrone 1990). Philolaus,
the third most famous Pythagorean after Pythagoras and Archytas, also
turns up as the author of several spurious treatises (Huffman 1993,
341 ff.), but a number of the forgeries were in the names of obscure
or otherwise unknown Pythagoreans. Thus, Callikratidas and Metopos are
presented as anticipating Plato's doctrine of the tripartite soul and
as using Plato's exact language to articulate it (Thesleff 1965, 103.5
and 118.1–4). Although there are indications that some ancient
scholars had doubts about the authenticity of the pseudo-Pythagorean
texts, for the most part they succeeded in their purpose all too well
and were accepted as genuine texts on which Plato and Aristotle
drew.

Although the pseudepigrapha are too varied to admit of one origin,
Centrone has recently argued that a core group of pseudepigrapha do
appear to be part of a single project (2014a). They are written in
Doric Greek (the dialect used in southern Italy where the Pythagoreans
flourished) in order to give them the appearance of authenticity and
share a common style. There are some twenty-five treatises belonging
to this group and they include some of the most famous pseudepigrapha,
including the work by ps.-Timaeus that was supposed to be Plato's
model, ps.-Archytas' works on categories and ps.-Ocellus On the
Universe. These treatises espouse the same basic system and seem
designed to cover all the basic fields of knowledge. The system is
based on theory of principles in which God is the supreme entity above
a pair of principles, one of which is limited and the other unlimited,
and which are identified with Aristotelian form and matter. This
system is very similar to what is found in Eudorus, a Platonist
working in Alexandria in the fist cenutury BCE. Starting from these
principles a common system is then developed which applies to
theology, cosmology, ethics, and politics. The connections to Eudorus
and to Philo who also worked in Alexandria, very much suggest that
this group of treatises was developed as a coherent project in
Alexandria sometime in the first century BCE or the first century
CE.

One important group of Pythagorean pseudepigrapha are those forged
in the names of Pythagorean women. Although some work has been done on
them there is still a pressing need for a comprehensive collection of
these texts and a study of them in light of the most recent
scholarship on Pythagoreanism. Pomeroy 2013 provides some useful
commentary but has serious drawbacks (see Centrone 2014b and Brodersen
2014). Many of the texts are collected in Thesleff 1965 under the
names Theano, Periktione, Melissa, Myia and Phintys and taken together
occupy about 15 pages of text. To Periktione are assigned two
fragments from a treatise On the Harmony of a
Woman. Periktione is the name of Plato's mother and it is
probable that hers is the famous name in which these works were
forged. Two further fragments from On Wisdom are also
assigned to her. These fragments show a strong similarity to fragments
from a treatise with identical title by Archytas and are likely to
have been assigned to Periktione by mistake (Huffman 2005, 598). Two
fragments from a work On the Temperance of a Woman are
assigned to Phintys. For Theano, the most famous Pythagorean woman
(see 3.3 above), one fragment of a work On Piety is preserved
as well as the titles of several other works, numerous apophthegms and
a number of letters. On Theano in the pseudepigraphal tradition see
Huizenga 2013, 96–117. Melissa and Myia are represented by one
letter each. With few exceptions the works focus on female virtue,
proper marital conduct, and practical issues such as how to choose a
wet nurse and how to deal with slaves. The advice is quite
conservative, stressing obedience to one's husband, chastity and
temperance. There is little that is specifically Pythagorean. Since
the authors are pseudonymous it is impossible to be sure whether they
were in fact written by women using female pseudonyms or men using
female pseudonyms (Huizenga 2013, 116). In the case of the letters
Städele's edition (1980) is to be preferred to Thesleff
(1965). The letters of Melissa and Myia along with three letters of
Theano are often found together in the manuscript tradition and may
have come to be seen as offering a curriculum for the moral training
of women (Huizenga 2013). Due to the dearth of preserved writings by
women from the ancient world some have been tempted to suppose that
the writings are genuine works by the named authors. However, as
demonstrated above, Pythagorean pseudepigrapha were very widespread
and more common than genuine Pythagorean works. In such a context the
onus of proof is on someone who wants to show that a work is
genuine. The content of the writings by Pythagorean women is simply
too general to make a convincing case that a specific writing could
only have been written by the supposed author rather than by a later
forger. In fact, the writings by women fit the pattern of the rest of
the pseudepigrapha very well. They are generally forged in the name of
famous Pythagorean women, whose names give authority to the advice
imparted (Huizenga 2013, 117). How better could one impart force to
advice to women than to assign that advice to women who belonged to
the philosophical school that gave most prominence to women? The
pseudepigrapha written in the names of Pythagorean women probably
mostly date to the first centuries BCE and CE like the other
Pythagorean pseudepigrapha, but certainty is not possible.

One of the most discussed treatises among the pseudepigrapha are
the Pythagorean Notes, which were excerpted by Alexander
Polyhistor in the first century BCE, who was in turn quoted by
Diogenes Laertius in his Life of Pythagoras (VIII
24–33). Thus the Notes date before the middle of the
first century BCE (probably towards the end of the third century BCE
[Burkert 1972a, 53]) and are earlier than most pseudepigrapha. In
Diogenes' life the Pythagorean Notes serve as the main
statement of Pythagoras' philosophical views. The treatise is wildly
eclectic, drawing from Plato's Timaeus, the early Academy and
Stocisim and the scholarly consensus is that the treatise is a forgery
(Burkert 1961, 26ff., Long 2013, Laks 2014). It is tempting to suppose
that some early material may be preserved amidst later material, but
the text is such an amalgam that it is in practice impossible to
identify securely any early material (Burkert 1961, 26; Laks 2014,
375). The Notes are well organized and present a complete if
compressed philosophy organized around the concept of purity (Laks
2014). Starting from basic principles (the Platonic monad and dyad)
they give an account of the world, living beings, and the soul ending
with moral precepts (some of the Pythagorean acusmata). Kahn
thought that the treatise reflected a Pythagorean community that was
active in the Hellenistic period (2001, 83) but Long is more likely to
be right that its learned eclecticism suggests that it is a scholarly
creation (Long 2013, 158–159).

“Neopythagorean” is a modern label, which overlaps with
two other modern labels, “Middle Platonist” and
“Neoplatonist,” so that a given figure will be called a
Neoplatonist or Middle Platonist by some scholars and a Neopythagorean
by others. There are several different strands in Neopythagoreanism.
One strand focuses on Pythagoras as a master metaphysician. In this
guise he is presented as the author of a theory of principles, which
went even beyond the principles of Plato's later metaphysics, the one
and the indefinite dyad, and which shows similarities to the
Neoplatonic system of Plotinus. The first Neopythagorean in this sense
is Eudorus of Alexandria, who was active in the middle and later part
of the first century BCE. He evidently presented his own innovations as
the work of the Pythagoreans (Dillon 1977, 119). According to Eudorus,
the Pythagoreans posited a single supreme principle, known as the one
and the supreme god, which is the cause of all things. Below this first
principle are a second one, which is also called the monad, and the
indefinite dyad. These latter two are Plato's principles in the
unwritten doctrines, but Eudorus says they are properly speaking
elements rather than principles (Simplicius, in Phys.,
CAG IX 181. 10–30). The system of principles described
by Eudorus also appears in the pseudo-Pythagorean writings (e.g.,
pseudo-Archytas, On Principles; Thesleff 1965, 19) and it is
hard to be certain in which direction the influence went (Dillon 1977,
120–121). On Eudorus' connection to the pseudo-Pythagorean
writings see also Bonazzi 2013 and Centrone 2014. A generation after
Eudorus, another Alexandrian, the Jewish thinker Philo, used a
Pythagorean theory of principles, which is similar to that found in
Eudorus, and Pythagorean number symbolism in order to give a
philosophical interpretation of the Old Testament (Kahn 2001,
99–104; Dillon 1977, 139–183). Philo's goal was to show
that Moses was the first philosopher. For Philo Pythagoras and his
travels to the east evidently played a crucial role in the
transmission of philosophy to the Greeks (Dillon 2014). Philo like
Eudorus has close connections to the Pythagorean pseudepigrapha
(Centrone 2014).

Moderatus of Gades (modern Cadiz in Spain), who was active in the
first century CE, shows similarities to Eudorus in his treatment of
Pythagorean principles. Plutarch explicitly labels him a Pythagorean
and presents his follower, Lucius, as living a life in accord with the
Pythagorean taboos, known as symbola or acusmata
(Table Talk 727b). It is thus tempting to assume that
Moderatus too lived a Pythagorean life (Dillon 1977, 345). His
philosophy is only preserved in reports of other thinkers, and it is
often difficult to distinguish what belongs to Moderatus from what
belongs to the source.

He wrote a comprehensive eleven volume work entitled Lectures on
Pythagoreanism from which Porphyry quotes in sections 48–53 of his
Life of Pythagoras. In this passage, Moderatus argues that the
Pythagoreans used numbers as a way to provide clear teaching about
bodiless forms and first principles, which cannot be expressed in
words. In another excerpt, he describes a Pythagorean system of
principles, which appears to be developed from the first two deductions
of the second half of Plato's Parmenides. In this system there
are three ones: the first one which is above being, a second one which
is identified with the forms and which is accompanied by intelligible
matter (i.e. the indefinite dyad) and a third one which is identified
with soul. The first two ones show connections to Eudorus'
account of Pythagorean first principles; the whole system anticipates
central ideas of the most important Neoplatonist, Plotinus (Dillon
1977, 346–351; Kahn 2001, 105–110).

Moderatus was a militant Neopythagorean, who explicitly charges that
Plato, Aristotle and members of the early academy claimed as their own
the most fruitful aspects of Pythagorean philosophy with only small
changes, leaving for the Pythagoreans only those doctrines that were
superficial, trivial and such as to bring discredit on the school
(Porphyry, VP 53). These trivial doctrines have been thought
to be the various taboos preserved in the symbola, but, since
his follower Lucius is explicitly said to follow the symbola,
it seems unlikely that Moderatus was critical of them. The charge of
plagiarism might suggest that Moderatus was familiar with the
pseudo-Pythagorean treatises, which appear to have been forged in part
to show that Pythagoras had anticipated the main ideas of Plato and
Aristotle (see Kahn 2001, 105).

It is with Numenius (see Dillon 1977, 361–379 and Kahn 2001,
118–133, and the entry on
Numenius, especially section 2), who
flourished ca. 150 CE in Apamea in northern Syria (although he may
have taught at Rome), that Neopythagoreanism has the clearest direct
contact with the great Neoplatonist, Plotinus. Porphyry reports that
Plotinus was, in fact, accused of having plagiarized from Numenius and
that, in response, Amelius, a devotee of Numenius' writings and
follower of Plotinus, wrote a treatise entitled Concerning the
Difference Between the Doctrines of Plotinus and Numenius
(Life of Plotinus 3 and 17). The third century Platonist,
Longinus, to a degree describes Plotinus himself as a Neopythagorean,
saying that Plotinus developed the exegesis of Pythagorean and
Platonic first principles more clearly than his predecessors, who are
identified as Numenius, his follower Cronius, Moderatus and
Thrasyllus, all Neopythagoreans (Porphyry, Life of Plotinus
20). Numenius also had considerable influence on Porphyry (Macris
2014, 396), Iamblichus (O'Meara 2014, 404–405) and Calcidius
(Hicks 2014, 429).

Numenius is regularly described as a Pythagorean by the sources that
cite his fragments such as Eusebius (e.g. Fr. 1, 4b, 5 etc. Des
Places). He presents himself as returning to the teaching of Plato and
the early Academy. That teaching is in turn presented as deriving from
Pythagoras. Plato is described as “not better than the great
Pythagoras but perhaps not inferior to him either” (Fr. 24 Des
Places). Strikingly, Numenius presents Socrates too as a Pythagorean,
who worshipped the three Pythagorean gods recognized by Numenius (see
below). Thus Plato derived his Pythagoreanism both from direct contact
with Pythagoreans and also from Socrates (Karamanolis 2006,
129–132). For Numenius a true philosopher adheres to the teaching of
his master, and he wrote a polemical treatise, directed particularly
at the skeptical New Academy, with the title On the Revolution of
the Academics against Plato (Fr. 24 Des Places). Numenius
presents the Pythagorean philosophy to which Plato adhered as
ultimately based on a still earlier philosophy, which can be found in
Eastern thinkers such as the Magi, Brahmans, Egyptian priests and the
Hebrews (Fr. 1 Des Places). Thus, Numenius was reported to have asked
“What else is Plato than Moses speaking Greek?” (Fr. 8 Des
Places).

Numenius presents his own doctrine of matter, which is clearly
developed out of Plato's Timaeus, as the work of Pythagoras
(Fr. 52 Des Places). Matter in its disorganized state is identified
with the indefinite dyad. Numenius argues that for Pythagoras the dyad
was a principle independent of the monad; later thinkers, who tried to
derive the dyad from the monad (he does not name names but Eudorus,
Moderatus and the Pythagorean system described by Alexander Polyhistor
fit the description), were thus departing from the original teaching.
In emphasizing that the monad and dyad are independent principles,
Numenius is indeed closer to the Pythagorean table of opposites
described by Aristotle and to Plato's unwritten doctrines. Since it is
in motion, disorganized matter must have a soul, so that the world and
the things in it have two souls, one evil derived from matter and one
good derived from reason. Numenius avoids complete dualism in that
reason does have ultimate dominion over matter, thus making the world
as good as possible, given the existence of the recalcitrant
matter.

The monad, which is opposed to the indefinite dyad, is just one of
three gods for Numenius (Fr. 11 Des Places), who here follows Moderatus
to a degree. The first god is equated with the good, is simple, at rest
and associates only with itself. The second god is the demiurge, who by
organizing matter divides himself so that a third god arises, who is
either identified with the organized cosmos or its animating principle,
the world soul (Dillon 1977, 366–372). Numenius is famous for the
striking images by means of which he elucidated his philosophy, such as
the comparison of the helmsman, who steers his ship by looking at the
heavens, to the demiurge, who steers matter by looking to the first god
(Fr. 18 Des Places). Numenius' argument that there is a first god
above the demiurge is paralleled by a passage in another treatise,
which shows connections to Neopythagorean metaphysics, The
Chaldaean Oracles (Majercik 1989), which were published by Julian
the Theurgist, during the reign of Marcus Aurelius (161–180 CE) and
thus at about the same time as Numenius was active. It is hard to know
which way the influence went (Dillon 1977, 363).

In The Refutation of all Heresies, the Christian bishop
Hippolytus (died ca. 235 CE) adopts the strategy of showing that
Christian heresies are in fact based on the mistaken views of pagan
philosophers. Hippolytus spends considerable time describing
Pythagoreanism, since he regards it as the primary source for gnostic
heresy (see Mansfeld 1992 for this and what follows). Hippolytus'
presentation of Pythagoreanism, which groups together Pythagoras,
Plato, Empedocles and Heraclitus into a Pythagorean succession,
belongs to a family of Neopythagorean interpretations of
Pythagoreanism developed in the first century BCE and the first two
centuries CE and which also appear in later commentators such as
Syrianus and Philoponus. Hippolytus' interpretation shows similarities
to material in Eudorus, Philo Judaeus, Plutarch and Numenius among
others, although he adapts the material to fit his own purposes. He
regards Platonism and Pythagoreanism as the same philosophy, which
ultimately derives from Egypt. Empedocles is regarded as a Pythagorean
and is quoted, sometimes without attribution, as evidence for
Pythagorean views. According to Hippolytus the Monad and the Dyad are
the two Pythagorean principles, although the Dyad is derived from the
Monad. The Pythagoreans recognize two worlds, the intelligible, which
has the Monad as its principle, and the sensible, whose principle is
the tetraktys, the first four numbers, which correspond to
the point, line, surface and solid. The tetraktys contains
the decad, since the sum of 1, 2, 3 and 4 is 10, and this is embodied
in the ten Aristotelian categories, which describe the sensible
world. The pseudo-Archytan treatise, The Whole System of
Categories, had already claimed this Aristotelian doctrine for
the Pythagoreans (see 4.2 above). Finally, the intelligible world is
equated with Empedocles' sphere controlled by the uniting power of
Love in contrast to the world of sense perception in which the
dividing power of Strife plays the role of the demiurge
(Refutation of all Heresies 6, 23–25).

A second strand of Neopythagoreanism, while maintaining connection
to these metaphysical speculations, emphasizes Pythagoras' role
in the mathematical sciences. Nicomachus of Gerasa (modern Jerash in
Jordan) was probably active a little before Numenius, in the first half
of the second century CE. Unlike Neopythagoreans such as Eudorus,
Moderatus and Numenius, whose works only survive in fragments, two
complete works of Nicomachus survive, Introduction to
Arithmetic and Handbook of Music. More than anyone else
in antiquity he was responsible for popularizing supposed Pythagorean
achievements in mathematics and the sciences. The Handbook of
Music gives the canonical but scientifically impossible story of
Pythagoras' discovery of the whole number ratios, which
correspond to the basic concordant intervals in music: the octave
(2:1), fifth (3:2), and fourth (4:3); he supposedly heard the concords
in the sounds produced by hammers of varying weights in a blacksmith's
shop, which he happened to be passing (Chapter 6 — translation in
Barker 1989, 256 ff.). In the next century, Iamblichus took this
chapter over virtually verbatim and without acknowledgement in his
On the Pythagorean Life (Chapter 26) and it was repeated in
many later authors. The harmonic theory presented by Nicomachus in the
Handbook is not original and is, in fact, somewhat
retrograde. It is tied to the diatonic scale used by Plato in
the Timaeus (35b-36b), which was previously used by the
Pythagorean Philolaus in the fifth-century (Fr. 6a Huffman) and shows
no awareness of or interest in the more sophisticated analysis of
Archytas in the fourth century BCE. Nicomachus is not concerned with
musical practice but with “what pure reasoning can reveal about
the properties of a rationally impeccable and unalterable system of
quantitative relations” (Barker 2007, 447). Nicomachus also
relies heavily and without acknowledgement on a non-Pythagorean
treatment of music, Aristoxenus'
Elementa Harmonica, many of the ideas of which he assigns to
the Pythagoreans (e.g., in Chapter 2; see Barker 1989, 245 ff.).

The Handbook was influential because it put forth an
accessible version of Pythagorean harmonics (Barker 2014,
200–202). Nicomachus provided a more detailed treatment of
Pythagorean harmonics in his lost
Introduction to Music. Most scholars agree that Books I-III
and perhaps Book IV of Boethius' De Institutione Musica are a
close paraphrase, which is often essentially a translation, of
Nicomachus' lost work (see Bower in Boethius 1989, xxviii and Barker
2007, 445). Even more influential than his work on harmonics was
Nicomachus' Introduction to Arithmetic. Again Nicomachus was
not an original or particularly talented mathematician, but this
popularizing textbook was widely influential. There were a series of
commentaries on it by Iamblichus (3rd CE), Asclepius of Tralles (6th
CE), and Philoponus (6th CE) and it was translated into Latin already
in the second half of the second century by Apuleius. Most
importantly, Boethius (5th-6th CE) provides what is virtually a
translation of it in his De Institutione Arithmetica, which
became the standard work on arithmetic in the middle ages. On
Boethius' use of Nicomachus see Hicks 2014, 422–424.

In the Introduction to Arithmetic, Nicomachus assigns to
Pythagoras the Platonic division between the intelligible and sensible
world, quoting the Timaeus as if it were a Pythagorean text (I
2). He also assigns Aristotelian ideas to Pythagoras, in particular a
doctrine of immaterial attributes with similarities to the Aristotelian
categories (I 1). Nicomachus divides reality into two forms, magnitude
and multitude. Wisdom is then knowledge of these two forms, which are
studied by the four sciences, which will later be known as the
quadrivium: arithmetic, music, geometry and astronomy. He
quotes a genuine fragment of Archytas (Fr. 1 Huffman) in support of the
special position of these four sciences. Nicomachus presents arithmetic
as the most important of the four, because it existed in the mind of
the creating god (the demiurge) as the plan which he followed in
ordering the cosmos (I 4), so that numbers thus appear to have replaced the
Platonic forms as the model of creation (on forms and numbers in Nicomachus see Helmig 2007). It is striking that,
along with this Platonization of Pythagoreanism, Nicomachus does give
an accurate presentation of Philolaus' basic metaphysical
principles, limiters and unlimiteds, before attempting to equate them
with the Platonic monad and dyad (II 18).

Another work by Nicomachus, The Theology of Arithmetic,
which can be reconstructed from a summary by Photius and an anonymous
work sometimes ascribed to Iamblichus and known as the
Theologoumena Arithmeticae (Dillon 1977, 352–353),
suggests that he largely returned to the system of principles found in
Plato's unwritten doctrines and did not follow Eudorus and Moderatus
in attempts to place a supreme god above the demiurge. Nicomachus
apparently presents the monad as the first principle and demiurge,
which then generates the dyad, but much is unclear (Dillon 1977,
353–358). The Theology of Arithmetic may have been most
influential in its attempt to set up an equivalence between the pagan
gods and the numbers in the decad, which was picked up later by
Iamblichus and Proclus (Kahn 2001, 116). Nicomachus also wrote a
Life of Pythagoras, which has not survived but which Porphyry
(e.g., VP 59) and Iamblichus used (Rohde 1871–1872;
O'Meara 2014, 412–413).

After Plotinus (205–270 CE), Neopythagoreanism becomes absorbed into
Neoplatonism. Although Plotinus was clearly influenced by
Neopythagorean speculation on first principles (see above), he was not
a Neopythagorean himself, in that he did not assign Pythagoras a
privileged place in the history of Greek philosophy. Plotinus treats
Pythagoras as just one among many predecessors, complains of the
obscurities of his thought and labels Plato and not Pythagoras as
divine (Enneads IV 8.11 ff.).

The earliest extant Life of Pythagoras is that of Diogenes
Laertius, who was active ca. 200 CE. The most recent treatment of
Diogenes' life is Laks 2014, on which much of what follows
depends. Unlike his successors Porphyry and Iamblichus (see below)
Diogenes had no philosophical affiliation and hence no philosophical
axe to grind in presenting the life of Pythagoras. Indeed, it is
striking that his life shows little influence from the Neopythagorean
authors discussed above. Diogenes draws on a wide variety of important
sources, some going back to the fourth century and others deriving
from the Hellenistic period. This material is put together in a very
loose, sometimes undetectable, organizational structure. There is a
notable section on Pythagoras' supposed writings (VIII, 6–7). He
shows particular interest in the Pythagorean way of life and quotes a
large number of Pythagorean symbola for some of which his
source was Aristotle (VIII 34–35). The main section on
Pythagoras' philosophical doctrines is a long quotation from the
first-century polymath Alexander Polyhistor who claims to be in turn
drawing on a treatise called Pythagorean Notes (VIII
24–33). For more on this treatise see the section on Pythagorean
pseudepigrapha above (4.2). Diogenes quotes a number of passages
satirizing Pythagoras, including Xenophanes' famous puppy fragment,
and presents some of his own epigrams making fun of the Pythagorean
way of life (VIII, 36). However, other parts of his life present
Pythagoras in a quite postive light so that it is hard to determine
precisely what attitude Diogenes took towards Pythagoras (Laks 2014,
377–380).

The Life of Pythagoras by Plotinus' pupil and editor,
Porphyry (234-ca. 305) is one of our most important sources for
Pythagoreanism (For what follows see Macris 2014). It was originally
part of his now lost Philosophical History. Continuing
interest in Pythagoras in later centuries led the Life of
Pythagoras to be preserved separately and it is the only large
section of the Philosophical History to
survive. The Philosophical History ended with Plato and
clearly regarded Platonic philosophy as the true philosophy so that
Pythagoras seems to have been highlighted as a key figure in the
development of Plato's philosophy. Porphyry's Life of
Pythagoras is particularly valuable, because he often clearly
identifies his sources. This same penchant for identifying and seeking
out important Pythagorean sources can be seen in his commentary on
Ptolemy's Harmonics (2nd CE), in which he preserves several
genuine fragments of the early Pythagorean Archytas, along with some
pseudo-Pythagorean material. In the Life of Pythagoras
Porphyry does not structure his information according to any
overarching theme but instead sets out the information derived from
other sources in a simple and orderly way with the minimum of
editorial intervention. Although he cites some fifteen sources, some
going back to the fourth century BCE, it is likely that he did not use
most of these sources but rather found them quoted in the four main
sources, which he used directly: 1) Nicomachus' Life of
Pythagoras, 2) Moderatus' Lectures on Pythagoreanism, 3)
Antonius Diogenes' novel Unbelievable Things Beyond Thule,
and 4) a handbook of some sort. Since these sources come from the
first and second centuries CE, Porphyry basically provides us with the
picture of Pythagoras common in Middle Platonism. This Pythagoras is
the prototype of the sage of old who was active as a teacher and tied
to religious mystery. However, he is not yet Iamblichus' priviliged
soul sent to save humanity (Macris, 2014, 390). Porphyry provides
little criticism of his sources and, although his life has a neutral
factual tone, in contrast to Diogenes Laertius in his Life of
Pythagoras, he includes no negative reports about Pythagoras.

It would appear, however, that Pythagoras was not made the source of
all Greek philosophy, but was rather presented as one of a number of
sages both Greek and non-Greek (e.g., Indians, Egyptians and Hebrews),
who promulgated a divinely revealed philosophy. This philosophy is, in
fact, Platonic in origin as it relies on the Platonic distinction
between the intelligible and sensible realms; Porphyry unhistorically
assigns it back to these earlier thinkers, including
Pythagoras. Pythagoras' philosophy is thus said to aim at freeing the
mind from the fetters of the body so that it can attain a vision of
the intelligible and eternal beings (Life of Pythagoras
46–47). O'Meara thus seems correct to conclude that Porphyry was
“…not a Pythagoreanizing Platonist … but rather a
universalizing Platonist: he finds his Platonism both in Pythagoras
and in very many other quarters” (1989, 25–29). Porphyry
himself lived an ascetic life that was probably largely inspired by
Pythagoreanism (Macris 2014, 393–394).

Porphyry's pupil, Iamblichus (ca. 245- ca. 325 CE), from Chalcis in
Syria, opposed his teacher on many issues in Neoplatonic philosophy and
was responsible for a systematic Pythagoreanization of Neoplatonism
(see O' Meara 1989 and 2014), particularly under the influence of
Nicomachus' earlier treatment of Pythagorean work in the
quadrivium. Iamblichus wrote a work in ten books entitled
On Pythagoreanism. The first four books have survived intact
and excerpts of Books V-VII are preserved by the Byzantine scholar
Michael Psellus. Book One, On the Pythagorean Life, has
biographical aspects but is primarily a detailed description of and a
protreptic for the Pythagorean way of life. It might be that
Iamblichus' Pythagoras is intended in part as a pagan rival to Christ
and to Christianity, which was gaining strength at this time.
Porphyry, indeed, had written a treatise Against the
Christians, now lost. In Iamblichus, Pythagoras' miraculous deeds
include a meeting at the beginning of his career with fishermen
hauling in a catch (VP 36; cf. Matthew 1. 16–20; see
Iamblichus, On the Pythagorean Life, Dillon and Hershbell
(eds.) 1991, 25–26). O'Meara, on the other hand, doubts this
connection to Christ (2014, 405 n. 21) and suggests that Iamblichus
may have constructed Pythagoras as a rival to Porphyry's presentation
of Plotinus as the model philosopher (1989, 214–215). In the end
we cannot be certain whether Iamblichus is responding to Porphyry or
Porphyry to Iamblichus, but they can be seen as battling over Plato's
legacy (O'Meara 2014, 403). Porphyry in his Life of Plotinus
and edition of his works is promoting Plotinus' interpretation of
Plato. Iamblichus, on the other hand, advocates a return to the
philosophy that inspired Plato, Pythagoreanism. Pythagorean philosophy
is portrayed by Iamblichus as a gift of the gods, which cannot be
comprehended without their aid; Pythagoras himself was sent down to
men to provide that aid (VP 1).

Iamblichus' On the Pythagorean Life is largely a compilation
of earlier sources but, unlike Porphyry, he does not usually identify
them. Rohde (1871-1872) argued influentially that On the
Pythagorean Life was largely a compilation from two sources:
Nicomachus' Life of Pythagoras and a life of Pythagoras by
Apollonius of Tyana. O'Meara argues that this underestimates both the
extent to which Iamblichus reworked his sources for his own
philosophical purposes and the variety of sources that he used
(O'Meara 2014, 412–415). A particularly clear example of
Iamblichus' distintive development of ideas found in earlier sources
can be seen in his treatment of the doctrine of the harmony of the
spheres (O'Meara 2007). It is also true that the remaining books
of On Pythgoreanism use a variety of sources. Book
Two, Protreptic to Philosophy, is an exhortation to
philosophy in general and to Pythagorean philosophy in particular and
relies heavily on Aristotle's lost Protrepticus. Book
Three, On General Mathematical Science, deals with the
general value of mathematics in aiding our comprehension of the
intelligible realm and is followed by a series of books on the
specific sciences. The treatment of arithmetic in Book IV takes the
form of a commentary on Nicomachus' Introduction to
Arithmetic. Books V-VII then dealt with arithmetic in physics,
ethics and theology respectively and were followed by treatments of
the other three sciences in the quadrivium: On Pythagorean
Geometry, On Pythagorean Music and On Pythagorean
Astronomy. Iamblichus was particularly interested in Pythagorean
numerology and his section on arithmetic in theology is probably
reflected in the anonymous treatise which has survived under the
title Theologoumena Arithmeticae and which has sometimes been
ascribed to Iamblichus himself. It appears that here again Iamblichus
relied heavily on Nicomachus, this time on his Theology of
Arithmetic.

It is possible that Iamblichus used the ten Books of On
Pythagoreanism as the basic text in his school, but we know that
he went beyond these books to the study of Aristotelian logic and the
Platonic dialogues, particularly the Timaeus and
Parmenides (Kahn 2001, 136–137). Nonetheless, it was because
of Iamblichus that Pythagoreanism in the form of numerology and
mathematics in general was emphasized by later Neoplatonists such as
Syrianus (fl. 430 CE) and Proclus (410/412–485 CE). Proclus is reported
to have dreamed that he was the reincarnation of Nicomachus (Marinus,
Life of Proclus 28). Proclus did treat Plato's writings as
clearer than the somewhat obscure writings of the Pythagoreans but his
Platonism is still heavily Pythagorean (O' Meara 2014, 415). The
successors of Proclus appear to follow his and Iamblichus'
interpretation of Pythagoras (O'Meara 2013).

A third strand in Neopythagoreanism emphasizes Pythagoras' practices
rather than his supposed metaphysical system. This Pythagoras is an
expert in religious and magical practices and/or a sage who lived the
ideal moral life, upon whom we should model our lives. This strand is
closely connected to the striking interest in and prominence of
Pythagoreanism in Roman literature during the first century BCE and
first century CE. Cicero (106–43 BCE) in particular refers to
Pythagoras and other Pythagoreans with some frequency. In De
Finibus (V 2), he presents himself as the excited tourist, who,
upon his arrival in Metapontum in S. Italy and even before going to
his lodgings, sought out the site where Pythagoras was supposed to
have died. At the beginning of Book IV (1–2) of the Tusculan
Disputations, Cicero notes that Pythagoras gained his fame in
southern Italy at just the same time that L. Brutus freed Rome from
the tyranny of the kings and founded the Republic; there is a clear
implication that Pythagorean ideas, which reached Rome from southern
Italy, had an influence on the early Roman Republic. Cicero goes on to
assert explicitly that many Roman usages were derived from the
Pythagoreans, although he does not give specifics. According to
Cicero, it was admiration for Pythagoras that led Romans to suppose,
without noticing the chronological impossibility, that the wisest of
the early Roman kings, Numa, who was supposed to have ruled from
715–673 BCE, had been a pupil of Pythagoras.

In addition to references to Pythagoras himself, Cicero refers to
the Pythagorean Archytas some eleven times, in particular emphasizing
his high moral character, as revealed in his refusal to punish in anger
and his suspicion of bodily pleasure (Rep. I 38. 59;
Sen. XII 39–41; Huffman 2005, 21–24, 283 ff. and 323 ff.).
Cicero's own philosophy is not much influenced by the Pythagoreans
except in The Dream of Scipio (Rep. VI 9), which owes
even more to Plato.

The interest in Pythagoras and Pythagoreans in the first century BCE
is not limited to Cicero, however. Both a famous ode of Horace (I 28
– see Huffman 2005, 19–21) and a brief reference in
Propertius (IV 1) present Archytas as a master astronomer. Most
striking of all is the speech assigned to Pythagoras that constitutes
half of Book XV of Ovid's
Metamorphoses (early years of the first century CE) and that
calls for strict vegetarianism in the context of the doctrine of
transmigration of souls. These latter themes are true to the earliest
evidence for Pythagoras, but the rest of Ovid's presentation assigns to
Pythagoras a doctrine that is derived from a number of early Greek
philosophers and in particular the doctrine of flux associated with
Heraclitus (Kahn 2001, 146–149).

This flourishing of Pythagoreanism in Roman literature of the golden
age has its roots in one of the earliest Roman literary figures,
Ennius (239–169 BCE), who, in his poem Annales, adopts
the Pythagorean doctrine of metempsychosis, in presenting himself as
the reincarnation of Homer, although he does not mention Pythagoras by
name in the surviving fragments. Roman nationalism also played a role
in the emphasis on Pythagoreanism at Rome. Since Pythagoras did his
work in Italy and Aristotle even referred to Pythagoreanism in some
places as the philosophy of the Italians
(e.g., Metaph. 987a10), it is not surprising that the Romans
wanted to emphasize their connections to Pythagoras. This is
particularly clear in Cicero's references to Pythagoreanism but once
again finds its roots even earlier. In 343 BCE during the war with the
Samnites, Apollo ordered the Romans to erect one statue of the wisest
and another of the bravest of the Greeks; their choice for the former
was Pythagoras and for the latter Alcibiades. Pliny, who reports the
story (Nat. XXXIV 26), expresses surprise that Socrates was
not chosen for the former, given that, according to Plato's
Apology, Apollo himself had labeled Socrates the wisest; it
is surely the Italian connection that explains the Romans' choice of
Pythagoras. Cicero (not Aristoxenus as suggested by Horky 2011)
connects the great wisdom assigned to the Samnite Herrenius Pontius to
his contact with the Pythagorean Archytas (On Old Age 41;
Huffman 2005, 329). This Roman attempt to forge a connection with
Pythagoras can also be seen in the report of Plutarch
(Aem. Paul. 1) that some writers traced the descent of the
Aemelii, one of Rome's leading families, to Pythagoras, by claiming
Pythagoras' son Mamercus as the founder of the house.

Although Rome's special connection to Pythagoras thus had earlier
roots, those roots alone do not explain the efflorescence of
Pythagoreanism in golden age Latin literature; some stimulus probably
came from the rebirth of what were seen as Pythagorean practices in the
way certain people lived. The two most learned figures in Rome of the
first century BCE, Nigidius Figulus and Varro, both have connections to
Pythagorean ritual practices. Thus we are told that Varro (116–27 BCE)
was buried according to the Pythagorean fashion in myrtle, olive and
black poplar leaves (Pliny, Nat. XXXV 160). Amongst Varro's
voluminous works was the Hebdomadês
(“Sevens”), a collection of 700 portraits of
famous men, in the introduction to which Varro engaged in praise for
the number 7, which is similar to the numerology of later
Neopythagorean works such as Nicomachus' Theology of
Arithmetic; in another work Varro presents a theory of gestation,
which has Pythagorean connections, in that it is based on the whole
number ratios that correspond to the concordant intervals in music
(Rawson 1985, 161).

It is Nigidius Figulus, praetor in 58, who died in exile in 45,
however, who is usually identified as the figure who was responsible
for reviving Pythagorean practices. In the preface to his translation
of Plato's Timaeus, which is often treated as virtually a
Pythagorean treatise by the Neopythagoreans, Cicero asserts of
Nigidius that “following on those noble Pythagoreans, whose
school of philosophy had to a certain degree died out, … this
man arose to revive it.” Some scholars are dubious about this
claim of Cicero. They point to the evidence cited above for the
importance of Pythagoreanism in Rome in the two centuries before
Nigidius and suggest that Cicero may be illegitimately following
Aristoxenus' claim that Pythagoreanism died out in the first half of
the fourth century (Riedweg 2005, 123–124). While there may be
some evidence that there were practicing Pythagoreans in the second
half of the fourth century (see above section 3.5), it is hard to find
anyone to whom to apply that label in the third and second centuries,
so that, from the perspective of the evidence available to us at
present, Cicero may well be right that Nigidius was the first person
in several centuries to claim to follow Pythagorean
practices. However, the sources for Nigidius are meager and there is
no evidence that he was the leader of a large and powerful group. If
there was an organized group at all, it is more likely to have been a
smaller circle (Flinterman 2014, 344).

It is difficult to be sure in what Nigidius' Pythagoreanism
consisted. There is no mention of Pythagoras or Pythagoreans in the
surviving fragments of his work nor do they show him engaging in
Pythagorean style numerology as Varro did (Rawson 1985, 291 ff.). In
Jerome's chronicle, Nigidius is labeled as Pythagorean and
magus; the most likely suggestion, thus, is that his
Pythagoreanism consisted in occult and magical practices. Pliny treats
Nigidius alongside the Magi and also presents Pythagoras and
Democritus as having learned magical practices from
the Magi. Cicero describes Nigidius as investgating matters
that nature had hidden and this may be a reference to such magical
lore (Flinterman 2014, 345). Nigidius' expertise as an astrologer (he
is reported to have used astrology to predict Augustus' future
greatness on the day of his birth [Suetonius, Aug. 94.5]) may
be another Pythagorean connection; Propertius' reference (IV 1) to
Archytas shows that Pythagorean work in astronomy was typically
connected to astrology in first century Rome.

What led Nigidius and Varro to resurrect purported Pythagorean cult
practices? One important influence may have been the Greek scholar
Alexander Polyhistor, who was born in Miletus but was captured by the
Romans during the Mithridatic wars and brought to Rome as a slave and
freed by Sulla in 80 BCE. He taught in Rome in the 70s. It is an
intriguing suggestion that Nigidius learned his Pythagoreanism from
Alexander (Dillon 1977, 117; For critiques of this suggestion see
Flinterman 2014, 349–350 and Long 2013, 145). There is no
evidence that Alexander himself followed Pythagorean practices, but he
wrote a book On Pythagorean Symbols, which was presumably an
account of the Pythagorean acusmata (or symbola),
which set out the taboos that governed many aspects of the Pythagorean
way of life. In addition, in his Successions of the
Philosophers, he gave a summary of Pythagorean philosophy, which
he supposedly found in the Pythagorean Notes (See section 4.2
above) and which has been preserved by Diogenes Laertius (VIII
25–35). The basic principles assigned to Pythagoras are those of
the Neopythagorean tradition that begins in the early Academy, i.e.,
the monad and the indefinite dyad. Since Alexander also assigns to the
Pythagoreans the doctrine that the elements change into one another,
we might suppose that Ovid also used Alexander directly or indirectly,
since he assigns a similar doctrine to Pythagoras in
the Metamorphoses (XV 75 ff., Rawson 1985, 294).

It is necessary to look in a slightly different direction, in order
to see how magical practices came to be particularly associated with
Pythagoras and thus why Nigidius was called Pythagorean and
magus. In the first century, it was widely believed that
Pythagoras had studied with the Magi (Cicero, Fin. V 87), i.e.
Persian priests/wise men. What Pythagoras was thought to have learned
from the Magi most of all were the magical properties of plants. Pliny
the elder (23–79 CE) identifies Pythagoras and Democritus as the
experts on such magic and the Magi as their teachers (Nat.
XXIV 156–160). Pliny goes on to give a number of specific examples from
a book on plants ascribed to Pythagoras. This book is universally
regarded as spurious by modern scholars, and even Pliny, who accepts
its authenticity, reports that some people ascribe it to Cleemporus. We
can date this treatise on plants to the first half of the second
century or earlier, since Cato the elder (234–149 BCE) appears to make
use of it in his On Agriculture (157), when he discusses the
medicinal virtues of a kind of cabbage, which was named after
Pythagoras (brassica Pythagorea).

A clearer understanding of this pseudo-Pythagorean treatise on
plants and a further indication of its date can be obtained by looking
at the work of Bolus of Mendes, an Egyptian educated in Greek (see
Dickie 2001, 117–122, to whom the following treatment of Bolus is
indebted). Bolus composed a work entitled Cheiromecta, which
means “things worked by hand” and may thus refer to potions
made by grinding plants and other substances (Dickie 2001, 119). Bolus
discussed not just the magical properties of plants but also those of
stones and animals. Pliny regarded the Cheiromecta as composed
by Democritus on the basis of his studies with the Magi (Nat.
24. 160) and normally cites its contents as what Democritus or the Magi
said. Columella, however, tells us what was really going on (On
Agriculture VII 5.17). The work was in fact composed by Bolus, who
published it under the name of Democritus. Bolus thus appears to have
made a collection of magical recipes, some of which do seem to have
connections to the Magi, since they are similar to recipes found in 8th
century cuneiform texts (Dickie 2001, 121). In order to gain authority
for this collection, he assigned it to the famous Democritus.

Since Democritus was sometimes regarded as the pupil of Pythagoreans
(Diogenes Laertius IX 38), Bolus' choice of Democritus to give
authority to his work may suggest that someone else (the Cleemporus
mentioned by Pliny?) had already used Pythagoras for this purpose and
that the pseudo-Pythagorean treatise on the magical properties of
plants was thus already in existence when Bolus wrote, in the first
half of the second century BCE. An example of the type of recipe
involved is Pliny's ascription to Democritus of the idea that the
tongue of a frog, cut out while the frog was still alive, if placed
above the heart of a sleeping woman, will cause her to give true
answers (Nat. XXXII 49). Thus, the picture of Pythagoras the
magician, which may lie behind a number of the supposed Pythagorean
practices of Nigidius Figulus, is based on little more than the
tradition that Pythagoras had traveled to Egypt and the east, so that
he became the authority figure, to whom the real collectors of magical
recipes in the third and second century BCE ascribed their
collections.

Nigidius' revival of supposed Pythagorean practices spread to
other figures in first century Rome. Cicero attacked Vatinius, consul
in 48 and a supporter of Caesar, for calling himself a Pythagorean and
trying to shield his scandalous practices under the name of Pythagoras
(Vat. 6). The scandalous practices involved necromancy,
invoking the dead, by murdering young boys. Presumably this method of
necromancy would not be ascribed to Pythagoras, but the suggestion is
that some methods of consulting the dead were regarded as Pythagorean.
Cicero later ended up defending this same Vatinius in a speech which
has not survived but some of the contents of which we know from the
ancient scholia on the speech against Vatinius. In this speech Cicero
defended Vatinius' habit of wearing a black toga, which he
attacked in the earlier speech (Vat. 12), as a harmless
affectation of Pythagoreanism (Dickie 2001, 170). Thus, the title of
Pythagorean in first century Rome carried with it associations with
magical practices, not all of which would have been widely
approved.

Another example of the connection between Pythagoreanism and magic
and its possible negative connotations is Anaxilaus of Larissa (Rawson
1985, 293; Dickie 2001, 172–173). In his chronicle, Jerome describes
him with the same words as he used for Nigidius, Pythagorean and
magus, and reports that he was exiled from Rome in 28 BCE. We
know that Anaxilaus wrote a work entitled Paignia
(“tricks”), which seems to have consisted of some rather
bizarre conjuring tricks for parties. Pliny reports one of
Anaxilaus' tricks as calling for burning the discharge from a
mare in heat in a flame, in order to cause the guests to see images of
horses' heads (Nat. XXVIII 181). The passion for things
Pythagorean can also be seen in the figure of king Juba of Mauretania
(ca. 46 BCE – 23 CE), a learned and cultured man, educated at Rome and
author of many books. Olympiodorus describes him as “a lover of
Pythagorean compositions” and suggests that Pythagorean books
were forged to satisfy the passion of collectors such as Juba
(Commentaria in Aristotelem Graeca 12.1, p. 13).

The connection between Pythagoreanism and astrology visible in
Nigidius can perhaps also be seen in Thrasyllus of Alexandria (d. 36
CE), the court astrologer and philosopher, whom the Roman emperor
Tiberius met in Rhodes and brought to Rome. Thrasyllus is famous for
his edition of Plato's dialogues arranged into tetralogies, but he was
a Platonist with strong Pythagorean leanings. Porphyry in his Life
of Plotinus (20) quotes Longinus as saying that Thrasyllus wrote
on Platonic and Pythagorean first principles (Dillon 1977,
184–185). Most suggestive of all is the quotation from
Thrasyllus preserved by Diogenes Laertius (Diogenes Laertius IX 38),
in which Thrasyllus calls Democritus a zealous follower of the
Pythagoreans and asserts that Democritus drew all his philosophy from
Pythagoras and would have been thought to have been his pupil, if
chronology did not prevent it. It is impossible to be sure what
Thrasyllus had in mind here, but one very plausible suggestion is that
he is thinking of Democritus as a sage, who practiced magic, the
Democritus created by Bolus, who was the successor to the arch mage
Pythagoras, the supposed author of the treatise on the magical uses of
plants (Dickie 2001, 195). Some have argued that the subterranean
basilica discovered near the Porta Maggiore and dating to the first
century CE was the meeting place of a Pythagorean community but the
evidence for this suggestion is very weak (Flinterman 2014).

We cannot be sure whether the Pythagoreanism of Nigidius, Varro and
their successors was limited to such things as burial ritual, magical
practices and black togas or whether it extended to less spectacular
features of a “Pythagorean” life. Q. Sextius, however,
founded a philosophical movement in the time of Augustus, which
prescribed a vegetarian diet and taught the doctrine of transmigration
of souls, although Sextius presented himself as using different
arguments than Pythagoras for vegetarianism (Seneca, Ep. 108.
17 ff.). One of these Sextians, as they were known, was Sotion, the
teacher of Seneca, and it is Seneca who gives us most of the
information we have about them. It is also noteworthy that Sextius is
also reported to have asked himself at the end of each day “What
bad habit have you cured today? What vice have you resisted? In what
way are you better” (Seneca, De Ira III 36). Cicero
tells us that it was “the Pythagorean custom” to call to
mind in the evening everything said, heard or done during the day
(Sen. 38, cf. Iamblichus, VP 164). The practice
described by Cicero is directed at training the memory in contrast to
Sextius' questions, which call for moral self-examination. On
Pythagoreanism in Rome see further Flinterman 2014.

Something similar to the Sextian version of the practice is found in
lines 40–44 of the Golden Verses, a pseudepigraphical
treatise consisting of 71 Greek hexameter verses, which were ascribed
to Pythagoras or the Pythagoreans. The poem is a combination of
materials from different dates, and it is uncertain when it took the
form preserved in manuscripts and called the Golden Verses;
dates ranging from 350 BCE to 400 CE have been suggested (see Thom
1995). It is not referred to by name until 200 CE. The Golden
Verses are frequently quoted in the first centuries CE and thus
constitute one model of the Pythagorean life in Neopythagoreanism, one
that is free from magical practices. Much of the advice is common to
all of Greek ethical thought (e.g., honoring the gods and parents;
mastering lust and anger; deliberating before acting, following
measure in all things), but there are also mentions of dietary
restrictions typical of early Pythagoreanism and the promise of
leaving the body behind to join the aither as an immortal.

Our most detailed account of a Neopythagorean living a life inspired
by Pythagoras is Philostratus' Life of Apollonius of
Tyana. Apollonius was active in the second half of the first
century CE and died in 97; Philostratus' life, which was written
over a century later at the request of the empress Julia Domna and
completed after her death in 217 CE, is more novel than sober
biography. According to Philostratus, Apollonius
identified his wisdom as that of Pythagoras, who taught him the proper
way to worship the gods, to wear linen rather than wool, to wear his
hair long, and to eat no animal food (I 32). Some have wondered if Apollonius' Pythagoreanism
is largely the creation of Philostratus, but the standard view has been that
Apollonius wrote a life of Pythagoras used by Iamblichus (VP
254) and Porphyry (Burkert 1972, 100), and the fragment of his treatise
On Sacrifices has clear connections to Neopythagorean
philosophy (Kahn 2001, 143–145). Rohde thought that large parts
of Apollonius's Life of Pythagoras could be found in
Iamblichus' On the Pythagorean Life, but recently more and
more doubt has arisen as to whether the Apollonius who wrote
the Life of Pythagoras used by Iamblichus is really
Apollonius of Tyana (Flinterman 2014, 357).

Like Pythagoras, Apollonius journeys to consult the wise men of the
east and learns from the Brahmins in India that the doctrine of
transmigration, which Apollonius inherited from Pythagoras, originated
in India and was handed on to the Egyptians from whom Pythagoras
derived it (III 19). Philostratus (I 2) emphasizes that Apollonius was
not a magician, thus trying to free him from the more disreputable
connotations of Pythagorean practices associated with figures such as
Anaxilaus and Vatinius (see above). Nonetheless, Philostratus' life
does recount a number of Apollonius' miracles, such as the raising of
a girl from the dead (IV 45). On Apollonius as a Pythagorean see
further Flinterman 2014.

These miracles made Apollonius into a pagan counterpart to Christ.
The emperor Alexander Severus (222–235 CE) worshipped Apollonius
alongside Christ, Abraham and Orpheus (Hist. Aug., Vita Alex.
Sev. 29.2). Hierocles, the Roman governor of Bithynia, who was
rigorous in his persecution of Christians, championed Apollonius at the
expense of Christ, in The Lover of Truth, and drew as a
response Eusebius' Against Hierocles. As mentioned
above, there is some probability that Iamblichus intends to elevate
Pythagoras himself as a pagan counterpart to Christ in his On the
Pythagorean Life (Dillon and Hershbell 1991, 25–26).

The satirist Lucian (2nd CE) provides us with a hostile portrayal of another
holy man with Pythagorean connections, Alexander of Abnoteichus in
Paphlagonia, who was active in the middle of the second century CE. In
Alexander the False Prophet, Lucian reports that Alexander
compared himself to Pythagoras (4), could remember his previous
incarnations (34) and had a golden thigh like Pythagoras (40). Lucian
shows the not often seen negative side to both Pythagoras' and
Alexander's reputations when he reports that, if one took even the
worst things said about Pythagoras, Alexander would far outdo him in
wickedness (4). Some have seen Alexander as largely a literary
construction by Lucian with little historical basis but other evidence
confirms that there were traveling Pythagorean wonder-workers in the
early imperial period (Flinterman 2014, 359).

Despite these attacks on figures such as Apollonius and Alexander
who modeled themselves on Pythagoras, the Pythagorean way of life was
in general praised; the Neopythagorean tradition which portrays
Pythagoras as living the ideal life on which we should model our own
reaches its culmination in Iamblichus' On the Pythagorean
Life and Porphyry's Life of Pythagoras

The influence of Pythagoreanism in the Middle Ages and Renaissance was
extensive and was found in most disciplines, in literature and art as
well as in philosophy and science. Here only the highlights of that
influence can be given (see further Heninger 1974, Celenza 1999,
Celenza 2001, Kahn 2001, Riedweg 2005, Hicks 2014 and Allen 2014, to
all of whom the following account is indebted). It is crucial to
recognize from the beginning that the Pythagoras of the Middle Ages
and Renaissance is the Pythagoras of the Neopythagorean tradition, in
which he is regarded as either the most important or one of the most
important philosophers in the Greek philosophical tradition. Thus,
Ralph Cudworth, in The True Intellectual System of the
Universe asserted that “Pythagoras was the most eminent of
all the ancient Philosophers” (1845, II 4). This is a far cry
from the Pythagoras that can be reconstructed by responsible
scholarship. Riedweg has put it well: “Had Pythagoras and his
teachings not been since the early Academy overwritten with Plato's
philosophy, and had this ‘palimpsest' not in the course of the
Roman empire achieved unchallenged authority among Platonists, it
would be scarcely conceivable that scholars from the Middle Ages and
modernity down to the present would have found the pre-Socratic
charismatic from Samos so fascinating” (2005, 128).

In the Middle Ages Pythagoras and Pythagorean philosophy were regarded
as the height of Greek philosophical achievement, although, somewhat
paradoxically Pythagoreanism was not still an active philosophy as
were Platonism and Aristotelianism but instead belonged to an
“imagined history” of philosophy (Hicks 2014, 420). The
view of Pythagoreanism in the Middle Ages was heavily determined by
three late ancient Latin writers: Calcidius, Macrobius and
Boethius. It was in particular the mathematical Pythagoreanism of
Nicomachus as transmitted by Boethius that determined the medieval
picture of Pythagoras. In ethics, Christians were able to embrace some
Pythagorean maxims such as the principle labeled Pythagorean by
Boethius: “Follow God” (Consolation of Philosophy
1.4). Some attention was also paid to other
Pythagorean symbola (see section 5.2 below). On the other
hand the doctrine of metempsychosis with its idea that human beings
were born again as animals was repugnant to Christian doctrine (John
of Salisbury, Policraticus 7.10). When it comes to
Pythagoras' life it is crucial to recognize that Iamblichus' and
Porphyry's lives of Pythagoras were not known in the Middle Ages so
that Pythagoras' activities were mostly known through passages from
classical authors and church fathers (Hicks 2014, 421). Pythagoras was
included in medieval encyclopedic works and was given particularly
thorough treatment by Vincent of Beauvais (before 1200–1264) in
his Speculum historiale (3.24–26), by John of Wales
(fl. 1260–1283) in Compendiloquium (3.6.2) and
in The Lives and Habits of the Philosophers ascribed to, but
probably not actually composed by, Walter Burley (1275–1344; see
Riedweg 2005, 129; Heninger 1974, 47; Hicks 2014, 421).

The most influential texts for the conception of Pythagoras in the
Latin Middle Ages and early Renaissance were Boethius' (480–524
CE) De Institutione Arithmetica and De Institutione
Musica, which are virtually translations of the Neopythagorean
Nicomachus' (second century CE) Introduction to Arithmetic
and Introduction to Music (this larger work is now lost, but
a smaller Handbook of Harmonics survives). Boethius followed
Nicomachus' classification of four mathematical sciences depending on
the nature of their objects (arithmetic deals with multitude in
itself, music with relative multitude, geometry with unmoving
magnitudes and astronomy with magnitude in motion). Boethius
introduced the term quadrivium, “fourfold road”
to understanding, to refer to these four sciences. In music theory,
Boethius presents the Pythagoreans as taking a middle position, which
gives a role in harmonics to both reason and perception. His
presentation of the Pythagorean position was central to music theory
for over a thousand years (Hicks 2014, 424). Boethius recounts the
apocryphal story of Pythagoras' discovery in a blacksmith's shop of
the ratios that govern the concordant intervals (Mus. I
10).

The medieval picture of Pythagoras as a natural philosopher and the
medieval understanding of his theory of the nature of the soul were
heavily influenced by the Latin commentary on Plato's Timaeus
by Calcidius (4th century CE) and the Commentary on the Dream of
Scipio by Macrobius (5th century CE). Calcidius regarded
Plato's Timaeus as a heavily Pythagorean document. Under the
influence of the Neopythagorean Numenius, Calcidius assigned to
Pythagoras the view that god was unity and matter duality (Hicks 2014,
429). Calcidius describes Plato's World-Soul in a way that highlights
its harmonic structure and Macrobius explicitly ascribes to Pythagoras
the view that the soul is a harmony (Commentary on the Dream of
Scipio 1.14.19). The doctrine of the harmony of the spheres,
which portrays the cosmos as a harmony that is expressed in the music
made by the revolutions of the planets, follows from the numerical
structure of the World-Soul and was also assigned to Pythagoras by
Calcidius. Most medieval Neoplatonic cosmoligies adopted the doctrine,
but the reintroduction of Aristotle's criticism of it in the
thirteenth century caused many to abandon the theory until it was
revived in the Renaissance by Ficino (Hicks 2014, 434). Later,
Shakespeare refers to the doctrine memorably in The Merchant of
Venice (V i. 54–65). Cicero's presentation of it in
the Dream of Scipio was also influential in the Renaissance
(Heninger 1974, 3).

Pythagorean influence also appeared at less elevated levels of
medieval culture. A fourteenth-century manual for preachers, which
contained lore about the natural world and is known as The Light
of the Soul, ascribes a series of odd observations about nature
to Archita Tharentinus, who is presumably intended to be the fourth
century BCE Pythagorean, Archytas of Tarentum. These are mostly cited
from a book, which was evidently forged in Archytas' name and known as
On Events in Nature. Some of the observations are plausible
enough, e.g., that a person at the bottom of a well sees stars in the
middle of the day, others more puzzling, e.g., that a dying man emits
fiery rays from his eyes at death, while still others may have
connections to magic, e.g., “if someone looks at a mirror,
before which a white flower has been placed, he cries.” Some
magical lore ascribed to an Architas is also found in the
thirteenth-century Marvels of the World (ps.-Albertus
Magnus), e.g., “if the wax of the left ear of a dog be taken and
hung on people with periodic fever, it is beneficial…”
(see Huffman 2005, 610–615). These texts seem to continue the
connection between Pythagoreanism and magic, which developed in the
third and second centuries BCE, and is prominent in Rome during the
first-century BCE (see above section 4.5).

In the Renaissance, Pythagoreanism played an important role in the
thought of fifteenth- and sixteenth century Italian and German
humanists. The Florentine Marsilio Ficino (1433–1499) is most
properly described as a Neoplatonist. He made the philosophy of Plato
available to the Latin-speaking west through his translation of all of
Plato into Latin. In addition he translated important works of
writers in the Neoplatonic and Neopythagorean tradition, such as
Plotinus, Porphyry, Iamblichus and Proclus. From that tradition he
accepted and developed the view that Plato was heir to an ancient
theology/philosophy (prisca theologia) that was derived from
earlier sages including Pythagoras, who immediately preceded Plato in
the succession (Allen 2014, 435–436). Ficino like the
Neopythagoreans had no conception of an early and a late
Pythagoreanism, for him Pythagoreanism was a unity as indeed was the
entire tradition of ancient theology (Celenza, 1999,
675–681). Ficino regarded works ascribed to the Chaldaean
Zoroaster, the Egyptian Hermes Trismegistus, Orpheus and Pythagoras,
which modern scholarship has shown to be forgeries of late antiquity,
as genuine works on which Plato drew (Kristeller 1979, 131). Ficino
provided a complete translation of the writings ascribed to Hermes
Trismegistus into Latin as well as translations of 39 of the short
Pythagorean sayings known as symbola, many of which are
ancient, and Hierocles' commentary on the pseudo-Pythagorean
Golden Verses (Heninger 1974, 63 and 66). The Golden
Verses (see Thom 1995) were, in fact, one of the most popular
Greek texts in the Renaissance and were commonly used in textbooks for
learning Greek; other pseudo-Pythagorean texts, such as the treatises
ascribed to Timaeus of Locri and Ocellus, were translated early and
regarded as genuine texts on which Plato drew (Heninger 1974, 49,
55–56). Ficino thought, moreover, that this whole pagan
tradition could be reconciled with Christian and Jewish religion and
accepted the view that Pythagoras was born of a Jewish father
(Heninger 1974, 201). For Ficino and the Renaissance as a whole,
Pythagoras was the most important of the Presocratic philosophers but
he never overshadowed Plato, who was the highest authority, in part
because there was no extensive body of texts by Pythagoras himself to
compete with the Platonic dialogues (Allen 2014, 453).

Ficino translated Iamblichus' four works on Pythagoreanism for his own
use and Iamblichus' On the Pythagorean Life had particular
influence on him. Ficino felt that in his time there was a need for a
divinely inspired guide on earth and fashioned himself as such a
prophet under the influence of Iamblichus' presentation of Pythagoras
as a divine guide sent by the gods to save mankind (Celenza 1999,
667–674). The Pythagorean musical practice that he found in
Iamblichus' On the Pythagorean Life , with its emphasis on
the impact of music on the soul, shaped his own music making and his
presentation of himself as a Pythagorean and Orphic holy man (Allen
2014, 436–440). Ficino and other Renaissance thinkers grappled
with the challenge that the Pythagorean notion
of metempsychosis presented to Christiantiy and how it might
be reconciled with Christian views (Allen 2014, 440–446). Ficino
was eager to absolve Plato from such a heresy. He does this in part by
treating metempsychosis metaphorically as referring to the soul's
ability to remake itself, but he also emphasized that metempsychosis
was not present in Plato's latest work, Laws, and made the
Pythagoreans scapegoats by suggesting that other passages in Plato
refer not to Plato's own doctrines but the Pythagoreans (Celenza 1999,
681–691). Ficino saw his own arithmology as Pythgorean and study
of Neopythagorean mathematical treatises by Nicomachus and Theon led
Ficino to conclude that Plato's nuptial number in Book 8 of
the Republic was 12 (Allen 2014, 446–450). He also
mistakenly and paradoxically followed the Neopythagoreans in thinking
that the Pythagoreans occupied the crucial position in the history of
philosophy of the first philosophers to distinguish between the
corporeal and incorporeal and to assert the superiority of the latter,
an achievement that is more reasonably assigned to Ficino's hero Plato
(Celenza 1999, 699–706).

The Pythagorean symbola were important to Ficino and the
Renaissance. They had already been interpreted as moral maxims by the
early church fathers (e.g., Clement, Origen and Ambrose). Ambrose, for
example, interpreted the Pythagorean "do not take the public path" to
mean that priests should live lives of exceptional purity
(Ep. 81). Jerome discussed 13 symbola in his Epistle
Against Rufinus and this list became the basis for medieval
discussions of the symbola in texts such as the Speculum
historiale of Vincent of Beauvais and the Lives and Habits of
the Philosophers of Walter Burley (Celenza 2001,
11–12). Ficino particularly encountered them in
Iamblichus' On the Pythagorean Life
and Protrepticus. For Ficino, their brevity was appropriate
to revealing the supreme reality, since he argued that the closer the
mind approaches to the One the fewer words it needs (Allen 2014,
450–451). In addition, he found them relevant to the preparation
and purification of the soul (Celenza, 1999, 693). They were widely
discussed by Ficino's contemporaries and successors (Celenza 2001,
52–81). Some figures wrote treatises devoted to their
interpretation (Ficino's mentor Antonio degli Agli, his follower
Giovanni Nesi [for an edition of Nesi's work see Celenza 2001],
Filippo Beroaldo the Elder and Lilio Gregorio Giraldi), while others
discussed them as part of larger works (Erasmus and Reuchlin). Not
everyone took the symbola seriously; Angelo Poliziano, the
great Florentine philologist and professor, presents a satire on them
in the fashion of Lucian, joking about Pythagoras' ability to talk to
animals and ridiculing the prohibition on beans (Celenza 2001,
33).

Ficino's friend and younger contemporary, Giovanni Pico della
Mirandola (1463–1494), advanced an even more radical doctrine of
universal truth, according to which all philosophies had a share of
truth and could be reconciled in a comprehensive philosophy (Kristeller
1979, 205). His Oration on the Dignity of Man shows the
variety of ways in which he was influenced by the Pythagorean
tradition. He equates the friendship that the Pythagoreans saw as the
goal of philosophy (see, e.g., Iamblichus, VP 229) with the
peace that the angels announced to men of good will (1965, 11–12); the
Pythagorean symbola forbidding urinating towards the sun or
cutting the nails during sacrifice are interpreted allegorically as
calling on us to relieve ourselves of excessive appetite for sensual
pleasures and to trim the pricks of anger (1965, 15); the practice of
philosophizing through numbers is assigned to Pythagoras along with
Philolaus, Plato and the early Platonists (1965, 25–26); Pythagoras is
said to have modeled his philosophy on the Orphic theology (1965, 33).
Finally, on the basis of the pseudo-Pythagorean letter of Lysis to
Hipparchus, Pythagoras is said to have kept silent about his doctrine
and left just a few things in writing to his daughter at his death. In
observing such silence, Pythagoras is portrayed as following an earlier
practice symbolized by the sphinx in Egypt and most of all by Moses,
who indeed published the law to men but supposedly kept the
interpretation of that law a secret. Pico equates this secret
interpretation of the law with the Cabala, an esoteric doctrine in
which the words and numbers of Hebrew scripture are interpreted
according to a mystical system (1965, 30; see also Heptaplus
1965, 68).

Pico's interest in reconciling the Cabala with Christianity and the
pagan philosophical tradition, including Pythagoreanism, was further
developed by the German humanist, Johannes Reuchlin (1445–1522). In the
dedicatory letter for his Three BooksOn the Art of the
Cabala (1517), which was addressed to Pope Leo X, Reuchlin says
that as Ficino has restored Plato for Italy so he will “offer to
the Germans Pythagoras reborn,” although he cannot “do this
without the cabala of the Hebrews, because the philosophy of Pythagoras
took its beginning from the precepts of the cabalists” (tr.
Heninger 1974, 245). Thus, in an earlier work (De verbo
mirifico) he had equated the four consonants in the Hebrew name
for God, JHVH, with the Pythagorean tetraktys, and gave to
each of the letters, which are equated with numbers as in Greek
practice, a mystical meaning. The first H, which also stands for the
number five that the Pythagoreans equated with marriage, is thus taken
to symbolize the marriage of the trinity with material nature, which
was equated with the dyad by the Neopythagoreans (Riedweg 2005,
130).

At the level of popular culture, several fortune-telling devices were
tied to Pythagoras, the most famous of which went under the name of
the Wheel of Pythagoras (Heninger 1974, 237). Pythagoras was probably
most widely known, however, through Ovid's presentation of him at the
beginning of Book XV of the Metamorphoses, which was
immensely popular in the Renaissance (Heninger 1974, 50). Ovid
recounts the story, which had already been recognized as apocryphal by
Cicero (Tusc. IV 1), that the second Roman king, Numa,
studied with Pythagoras. Pythagoras is presented inaccurately by Ovid
as a great natural philosopher, who discovered the secrets of the
universe and who believed in a doctrine of the flux of four
elements. On the other hand, Ovid's emphasis on the prohibition on
eating animal flesh and on the immortality of the soul have some
connection to the historical Pythagoras. In the Renaissance,
Pythagoras was not primarily known for the “Pythagorean
Theorem,” as he is today. Better known was the doubtful anecdote
(Burkert 1960, Riedweg 2005, 90–97), going back ultimately to
Heraclides of Pontus but known to the Renaissance mainly through
Cicero (Tusc. V 3–4), that he was the first to coin the word
“philosopher” (Heninger 1974, 29).

In the sixteenth century, Pythagorean influence was particularly
important in the development of astronomy. The Polish astronomer
Copernicus (1473–1543), in the Preface and Dedication to Pope Paul
III attached to his epoch making work, On the Revolution of
the Heavenly Spheres, reports that, in his dissatisfaction with
the commonly accepted geocentric astronomical system of Ptolemy (2nd
century CE), he laboriously reread the works of all the philosophers to
see if any had ever proposed a different system. This labor led him to
find inspiration not from Pythagoras himself but rather from later
Pythagoreans and in particular from Philolaus. Copernicus found in
Cicero (Ac. II 39. 123) that the Pythagorean Hicetas (4th
century BCE — Copernicus mistakenly calls him Nicetas) had proposed that
the earth revolved around its axis at the center of the universe and in
pseudo-Plutarch (Diels 1958, 378) that another Pythagorean, Ecphantus,
and Heraclides of Pontus (both 4th century BCE), whom Copernicus
regarded as a Pythagorean, had proposed a similar view. More
importantly, he also found in pseudo-Plutarch that the Pythagorean,
Philolaus of Croton (5th century BCE), “held that the earth moved
in a circle … and was one of the planets” (On the
Revolutions of the Heavenly Spheres 1. 5, tr. Wallis).

Copernicus reports to the Pope that he was led by these earlier
thinkers “to meditate on the mobility of the earth.”
Pythagorean influence on Copernicus was not limited to the notion of a
moving earth. In the same preface he explains his hesitation to publish
his book in light of the pseudo-Pythagorean letter of Lysis to
Hipparchus, which recounts the supposed reluctance of the Pythagoreans
to divulge their views to the common run of people, who had not devoted
themselves to study (for further Pythagorean influences on Copernicus
see Kahn 2001, 159–161). A number of the followers of Copernicus saw
him as primarily reviving the ancient Pythagorean system rather than
presenting anything new (Heninger 1974, 130 and 144, n. 131); Edward
Sherburne reflects the common view of the late 17th century in
referring to the heliocentric system as “the system of Philolaus
and Copernicus” (Heninger 1974, 129–130), although in the
Philolaic system it is, in fact, a central fire and not the sun that is
at the center of the universe.

The last great Pythagorean was Johannes Kepler (1571–1630 — see Kahn
2001, 161–172 for a good brief account of Kepler's Pythagoreanism).
Kepler began by developing the Copernican system in light of the five
regular solids (tetrahedron, cube, octahedron, dodecahedron and
icosahedron), to which Plato appealed in his construction of matter in
the Timaeus (see especially 53B-55C). He followed the
Renaissance practice illustrated above of regarding Greek philosophy as
closely connected to the wisdom of the Near East, when he asserted that
the Timaeus was a commentary on the first chapter of
Genesis (Kahn 2001, 162). In the preface to his early work,
Mysterium Cosmographicum (1596), Kepler says that his purpose
is to show that God used the five regular bodies, “which have
been most celebrated from the time of Pythagoras and Plato,” as
his model in constructing the universe and that “he accommodated
the number of heavenly spheres, their proportions, and the system of
their motions” to these five regular solids (tr. Heninger 1974,
110–111).

In ascribing geometrical knowledge of the five regular solids to
Pythagoras, Kepler is following an erroneous Neopythagorean tradition,
although the dodecahedron may have served as an early Pythagorean
symbol (see on Hippasus in section 3.4 above and
Burkert 1972, 70–71, 404, 460). Thus, this aspect of Kepler's work is
more Platonic than Pythagorean. The five solids were conceived of as
circumscribing and inscribed in the spheres of the orbits of the
planets, so that the five solids corresponded to the six planets known
to Kepler (Saturn, Jupiter, Mars, Earth, Venus, Mercury). There were
six planets, because there were precisely five regular bodies to be
used in constructing the universe, corresponding to the five intervals
between the planets. This view was overthrown by the later discovery of
Uranus as a seventh planet. Kepler's cosmology was, however, far from a
purely a priori exercise. Whereas his contemporary, Robert
Fludd, developed a cosmology structured by musical numbers, which could
in no way be confirmed by observation, Kepler strove to make his system
consistent with precise observations. Kahn suggests that we here see
again the split “between a rational and an obscurantist version
of Pythagorean thought,” which is similar to the ancient split in
the school between mathematici and acusmatici (2001,
163).

Close work with observational data collected by Tycho Brahe led
Kepler to abandon the universal ancient view that the orbits of the
planets were circular and to recognize their elliptical nature. More
clearly Pythagorean is Kepler's consistent belief that the data show
that the motions of the planets correspond in various ways to the
ratios governing the musical concords (see Dreyer 1953, 405–410), so
that there is a heavenly music, a doctrine attested for Philolaus and
Archytas, which probably goes back to Pythagoras as well (Huffman 1993,
279 ff.; Huffman 2005, 137 ff.). For Kepler, however, the music
produced by the heavenly motions was “perceived by reason, and
not expressed in sound” (Harmonice Mundi V 7). In his
attempt to make the numbers of the heavenly music work, he joked that
he would appeal to the shade of Pythagoras for aid, “unless the
soul of Pythagoras has migrated into mine” (Koestler 1959,
277).

Kepler has been described “as the last exponent of a form of
mathematical cosmology that can be traced back to the shadowy figure of
Pythagoras” (Field 1988, 170). It is true that Kepler's work led
the way to Newton's mechanics, which cannot be described in terms of
ancient geometry and number theory but relies on the calculus and which
relies on a theory of physical forces that is alien to ancient thought.
On the other hand, many modern scientists accept the basic tenet that
knowledge of the natural world is to be expressed in mathematical
formulae, which is rightly regarded as a central Pythagorean thesis,
since it was first rigorously formulated by the Pythagoreans Philolaus
( Fr. 4 — see Huffman 1993) and Archytas (Huffman 2005, 65 ff.) and
may, in a rudimentary form, go back to Pythagoras himself.

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